A Stochastic Linearized Augmented Lagrangian Method for Decentralized Bilevel Optimization
Anonymous Author(s)
AffiliationAddress email
Abstract
Bilevel optimization has been shown to be a powerful framework for formulatingmulti-task machine learning problems, e.g., reinforcement learning (RL) and metalearning, where the decision variables are coupled in both levels of the minimizationproblems.
In practice, the learning tasks would be located at different computingresource environments, and thus there is a need for deploying a decentralizedtraining framework to implement multi-agent and multi-task learning.
We developa stochastic linearized augmented Lagrangian method (SLAM) for solving generalnonconvex bilevel optimization problems over a graph, where both upper and loweroptimization variables are able to achieve a consensus.
We also establish that thetheoretical convergence rate of the proposed SLAM to the Karush-Kuhn-Tucker(KKT) points of this class of problems is on the same order as the one achieved bythe classical distributed stochastic gradient descent for only single-level nonconvexminimization problems.
Numerical results tested on multi-agent RL problemsshowcase the superiority of SLAM compared with the state-of-the-art methods.
Introduction
In this paper, we consider the following general decentralized bilevel optimization (DBO) frameworkwith applications to machine learning problems.
Suppose that there are n nodes over a connectedgraph G = {E , V} where E and V represent the edges and vertices.
Let N i denote the set ofneighboring nodes for node i .
Then the goal of DBO is to have these nodes jointly minimize twolevels of optimization problems.
More formally, DBO is expressed as
[Equation]
(1a)
[Equation]
1b)
[Equation]
,k n n (cid:88) i =
g i,k ( x i , y i,k ) s.t.
y i,k = y j,k , j ∈ N i , ∀ k ∈ [ m ] , (1c) where vector x i is the upper level (UL) optimization variable at each node i , vector y i,k denotesthe lower level (LL) decision variable for the k th learning task at node i , f i (; ) is a (smooth) ULloss function and possibly nonconvex with respect to (w.r.t.)
both the UL and LL variables, g i,k ( , )denotes the LL objective function of the k th task at node i , m represents the total number of LLoptimization problems, the consensus constraints x i =
y j , y i,k = y j,k , j ∈ N i , ∀ i ∈ [ n ] , ∀ k ∈ [ m ] ,enforce the model agreements at each level of the problems and for each LL learning task, and y ∗ k =[
y ∗ 1 ,k , . .
, y ∗ n,k ] T is the optimal solutions of the k th LL problem under the consensus constraints.
Applications of Bilevel Optimization.
Many machine learning problems can be formulated mathematically as a form of bilevel optimization or, more precisely, a special case of problem (1), e.g., meta-learning or meta reinforcement learning (RL), actor-critic (AC) schemes in RL, hyperparameteroptimization (HPO), and so on.
Meta-learning : Classical bilevel optimization is referred to as thecase where there is no consensus constraint but with only two levels of the minimization subproblems,i.e., min x f ( x , y ∗ ( x )) , s.t.
y ∗ ( x ) =
arg min y g ( x , y ) , which is also known as Stackelberg games[1] with the UL decision variable as the leader and the LL decision variable as the follower.
It turnsout that this class of optimization problems is useful in formulating a wide range of hierarchical ornested structured machine learning problems.
For example, one of the most popular domain adaptionlearning models, model-agnostic meta-learning (MAML) [2, 3], can be written as a special caseof bilevel programming [4], where the UL model provides a good initialization for acceleratinglearning procedures by implementing the LL algorithms.
The idea behind the model design is thatthe UL model is considered as the meta learner that searches for a permutation-invariant subspaceover multiple task-specific learners at the LL so that the performance of the MAML model can begeneralized well for unseen or testing data samples.
The theoretical analysis of the generalizationperformance of this class of bilevel problems has shown that MAML can indeed decrease the generalization error as the number of tasks increases, at least for strongly convex loss functions [5].
Subsequently, a thorough ablation study from the latent representation perspective shows that featurereuse is the actual dominant factor in improving the generalization performance of MAML [6], andthe authors propose a neural network-oriented algorithm with almost no inner loop (ANIL) thatsplits the neural network parameters into two parts corresponding to the UL and LL optimizationproblems, respectively.
Extensive numerical experiments illustrate that ANIL achieves almost thesame accuracy as the classical MAML but with significant computational savings.
This examplefurther strengthens the necessity of variable splitting in the learning structure by optimizing twolevels of objective functions to enhance the generalization performance.
Beyond the traditional meta-learning scenarios, MAML has also been applied to increasing thegeneralization ability of agents in RL problems by replacing the (stochastic) gradient with the(natural) policy gradient (PG) [3] under the same two-level structure.
RL : Besides meta-learningproblems, AC structure in RL is another class of common learning frameworks that can be formulatedby a bilevel optimization problem in nature [7, 8, 9], where the actor step at the UL aims at optimizingthe policy while the critic step at the LL is responsible for value function evaluation.
HPO : Besides,as the expressiveness of neural networks increased sharply over the past decades, the reuse of largemodels with adaptation to multi-task learning problems presents promising solutions by leveraging thepre-train and fine-tune strategy, such as in applications of HPO [10, 11] where the hyperparametersare trained at the UL problem so that the downstream learning tasks are learned with low costsincluding the expense of both computation and memory.
Applications of Multi-agent Settings.
When multiple computational resources are available andconnected, it is well motivated that exploring them solves distributed large-scale problems witha reduced amount of training time or performs multi-task learning.
The bilevel structure of themeta-learning (RL) is a good fit in this scenario as either UL/LL or both levels may need to accessthe networked data samples rather than local ones.
For example, a federated learning setting of
MAML [12] and bilevel optimization [13] have been built up over multiple nodes recently, where themeta/UL learner finds an initial shared model while the local/LL learners leverage it for adapting datadistributions of individual users.
In such a way, the federated MAML model can realize personalizedlearning without sharing heterogeneous data over numerous clients.
Once there is no central controllerfor coordinating the model aggregation, a diffusion-based MAML (Dif-MAML) [14] is proposedby spreading the model parameters over a network, where the UL parameter is updated by one stepof stochastic gradient descent (SGD) based on a combination of the parameters of neighbors as theinitialization for local model updates.
Decentralized hierarchical structured learning is even more stringent in the multi-agent RL (MARL)setting [15] as the learning tasks are essentially located at scattered sensors and/or controllers.
Specifically, in a fully collaborative setting, the team-based value function is required to be sharedover all the agents such that each agent is able to improve its policy based on the estimated totalreward.
For example, the decentralized AC (DAC) scheme has been investigated widely [15, 16, 17],where each agent uses the actor step to optimize its policy while the critic step performs one step[18] or multiple steps of temporal difference learning with mini-batch sampling (MDAC) [17, 19]and communications so that the team-based reward over the network is obtained by each agent.
Itturns out that DAC can be formulated as a special case of problem (1) as there is no consensus atthe UL.
Recently, it has been revealed that if there exists homogeneity of the state and action spaces, decentralized policy consensus (or a partial policy parameter sharing strategy) provides significantmerits to the centralized training and decentralized execution paradigm in terms of learning scalabilityand efficiency[18, 20], which motivates the consensus process at both UL and LL DBO problems.
Related Theoretical Works.
Given the fruitful results across these many applications, the corresponding theoretical analysis has been developing very fast as well for variants of bilevel optimizationproblems.
For example, the convergence behaviors of classical inexact MAML (iMAML) methodshave been quantified for both convex [21, 22] and nonconvex [23] cases of the UL loss function,where the LL algorithm only performs one step of stochastic gradient descent (SGD) based on the LLobjective functions as the adaptation step.
Moreover, the iteration complexity of ANIL with multipleiterations for minimizing the LL problems have been studied in [24], which justifies the significantcomputational advantages of ANIL compared with MAML in theory.
Furthermore, the finite-timeanalysis of AC algorithms has shown [25] that, once the learning rates at both the actor and critic sidesare chosen properly, a two timescale AC algorithm can achieve an O ( ϵ − 2 . 5 ) iteration complexity forfinding the first-order stationary points (FOSPs) of general nonconvex reward functions.
Besides these theoretical analyses in a specific learning setting, the algorithm design and corresponding convergence analysis for general bilevel optimization solvers have been recently advancing at arapid pace under certain assumptions that the UL objective function is general nonconvex while the LLobjective functions are strongly convex, which covers the existing convergence results shown for ACalgorithms.
The typical algorithms include those with double-loop structure, those with two timescaleor single timescale but single-loop, and those with error-correction or accelerated/variance-reduction.
To be more specific, double-loop algorithms, such as bilevel stochastic approximation (BSA) methods[26] and stochastic bilevel optimizers (stoBiO) [27], mainly request an inner loop to solve the LLproblem up to a certain error tolerance or with a certain number of iterations and then switch backto optimize the UL problem, which can achieve an O ( ϵ − 2 ) convergence rate to the ϵ -FOSPs.
Inpractice, single-loop algorithms are implemented more efficiently in terms of computational complexity and hyperparameter tuning compared to double-loop algorithms.
A two-timescale stochasticapproximation (TTSA) was analyzed in [28], but it is shown that TTSA needs O ( ϵ − 2 . 5 ) number ofiterations to achieve the ϵ -FOSPs.
Later, an error correction method, named the Single-TimescalestochAstic BiLevEl optimization (STABLE) method [29], improves the convergence rate of thesingle-loop algorithm to O
( ϵ − 2 ) and a tighter analysis for ALternating Stochastic gradient dEscenT(ALSET) shows that the single-loop algorithm can also achieve a convergence of O ( ϵ − 2 ) without theerror correction technique.
When more advanced momentum-assisted or variance reduction methodsare adopted in the algorithm design, the subsequent works, such as the momentum-based recursivebilevel optimizer (MRBO) [30] and the single-timescale double-momentum stochastic approximation(SUSTAIN) [31] and the variance reduced BiAdam (VR-BiAdam) [32], can sharpen the convergencerate of bilevel algorithms to O
( ϵ − 1 . 5 ) .
For the theoretical works on MAML/MARL, it is shown in [17, 19] that when the critic side isallowed the consensus step at each agent to approximate the networked rewards, MDAC algorithmscan achieve an O ( ϵ − 2 ) convergence rate to FOSPs, but both of them require an inner loop procedurefor the LL problem which makes the algorithms double loop.
Dif-MAML [14] is able to performthe UL consensus-based meta learning, but iMAML considered in Dif-MAML is only a very specialcase of bilevel.
Thus, the applicability of Dif-MAML is restrictive.
One of the closest works toours is CAC [18], which can realize the consensus on both UL and LL problems with O ( ϵ − 2 . 5 )number of iterations and is only for DAC problems.
A theoretical comparison between our workand closely related previous works on bilevel programming is shown in Table 1.
There is a lineof independent work on decentralized optimization.
But the existing works are only suitable forsingle-level minimization of only nonconvex problems, such as distributed SGD [33, 34], stochasticgradient tracking [35, 36] and stochastic primal dual algorithm [37], which can achieve an O (1 / ( nϵ 2 ))convergence rate to FOSPs for general nonconvex objective function optimization problems.
Main Contributions of This Work.
In this work, we consider a very general DBO setting, whereboth UL and LL problems can include a consensus constraint for model parameter sharing and therewould be multiple LL problems coupled with the UL problem.
To solve this problem efficiently in afully decentralized way, we propose a S tochastic L inearized A ugmented Lagrangian M ethod (SLAM)for dealing with both of the two levels of the optimization processes and the consensus constraints ateach level.
Leveraging the linearized augmented Lagrangian function as a surrogate, the design of
SLAM is simple and easily implemented as it is a single-loop algorithm with only step sizes to betuned for convergence.
We make the standard assumptions on Lipschitz continuity and convexity [Table] whether algorithm only needs one round of either or LL
[Table]
for both the UL and LL optimization problems as shown in the existing literature.
We establish theconditions of SLAM w.r
[Table] t. convergence to ϵ -Karush-Kuhn-Tucker (KKT) points of problem (1) at arate of O (1 / ( nϵ 2 )) , matching the standard convergence rate achieved by decentralized SGD type ofalgorithm to FOSPs for only single-level nonconvex minimization problems.
Remarkably, throughnumerical experiments on MARL problems, it is observed that SLAM can converge faster than theexisting MARL methods and even achieve higher rewards in some cases.
To summarize, the main contributions of this work are highlighted as follows:
▶ Our proposed SLAM algorithm is generic, and thus generalizes the single agent-based bilevelalgorithms to the multi-agent setting and is amnable to be specialized to solve multiple consensusbased DBO problems.
153 ▶ SLAM is a single-timescale and single-loop algorithm that can find the ϵ -KKT points at a rateof O (1 / ( nϵ 2 )) , which shows a linear speedup w.r
t. the number of nodes.
To the best of ourknowledge, this is the first work that shows a decentralized stochastic algorithm can achievethis rate under the constraints where any level or both levels of the DBO problem requires theconsensus process.
158 ▶ Numerical results that illustrate the proposed SLAM outperforms the state-of-the-art MARLalgorithms over heterogeneous networks in terms of both convergence speed and achievablerewards.
Due to space limitations, all technical proofs are deferred to the supplement.
Decentralized Bilevel Optimization Framework
Problem formulation of DBO.
One of the main motivations for performing decentralized jointlearning is dealing with large-scale dataset or scattered data samples.
At each node, the loss functioncan be written as f i ( x i , y ∗ 1 ( x i ) , . . .
, y ∗ m ( x i )) ≜
E ξ ∈D U i [ F i ( x i , y ∗ 1 ( x i ) , . . .
, y ∗ m ( x i ); ξ )] , where D U idenotes the local data distributions at the UL optimization problem, and F i () represents the estimationerror of the UL learning model on data ξ ∈ D U i .
Similarly, the LL learning tasks also include randomlysampled data from a local distribution D L i,k for task k , so the LL cost function at each node canbe expressed as g i,k
( x i , y i,k ) ≜
E ζ ∈D L i,k
[ G i,k ( x i , y i ; ζ )] , ∀ k , where G i,k denotes the estimationerror of the LL learning model over y k,i on data ζ ∈ D L i,k .
It is well known that SGD is one of themost efficient algorithms for tackling large amounts of data samples.
Before showing the algorithmdesign, we first reformulate problem (1) in a concise and compact way from a global view of thevariables.
Let x ≜ [ x 1 , . . .
, x n ] T and y k ≜
[ y 1 ,k , . . .
, y n,k ] T .
Then, problem (1) can be rewritten byconcatenated variables as
[Equation]
where A ∈ R |E|× n represents the incidence matrix 1 and f i ( x i , y ∗ i,k
( x i )) abbreviatesf i ( x i , y ∗ 1 ( x i ) , . . .
, y ∗ m ( x i )) for notational brevity.
Algorithm Design.
Towards this end, it is straightforward to construct a variant of the classicalaugmented Lagrangian function for the UL optimization problem as
[Equation]
where λ denotes the dual variable (Lagrangian multiplier) for the consensus constraint, ρ > 0 , and γis a scaling factor (which will be determined later).
Motivated by the primal-dual optimization framework [39], one step of gradient descent based on thelinearized objective function with a following gradient ascent step is sufficient for the minimizationof the general nonconvex loss function under the linear constraints, which means that there is no needto solve an inner optimization problem before updating the Lagrangian multiplier as is done in theclassical augmented Lagrangian method.
Note that given y ∗ i,k , when both the UL and LL objective functions are differentiable and the inverseof the Hessian matrix at the LL problem exists, i.e., ∇
y
i,k y i,k g i,k ( x i , y ∗ i,k ( x i )) is invertible, thenthere exists a closed form for ∇ f i ( x i , y ∗ i,k ( x i ))
due to the finite sum structure of this class of DBOproblems (please see Lemma 1 in the supplement for details).
Following the existing works on bilevelalgorithm designs, replacing y ∗ i,k ( x i ) by y i,k in ∇ f i ( x i , y ∗ i,k ( x i )) can serve as an efficient surrogatefor the stochastic gradient estimate in the sense that the dissimilarity between ∇ f i ( x i , y ∗ i,k
( x i ))
and∇
f i ( x i , y i,k ) shrinks exponentially as the number of i.i.d.
samples data increases.
Let h rg,k and h rfrespectively denote the stochastic unbiased gradient estimate of the LL and UL objective functionsg k ( x r , y rk ) , f ( x r , y r +1 k ) , ∀ k , w.r.t.
y k and x , where r represents the index of iterations.
Thus, ourproposed SLAM can be expressed as
[Equation]
where ω k is the dual variable for ensuring the LL consensus process for each learning task, α and βare the parameters of the quadratic penalization terms, and ρ/γ here is the step-size for the updatesof the dual variables.
Implementation of SLAM.
Noting that the objective functions in each subproblem, i.e., (4a) and(4c), are quadratic, we can easily have the updates of both UL and LL optimization variables as
[Equation]
where 1 /α and 1 /β serve as the step-sizes of updating both UL and LL learning models.
Subtracting the equality with the same one from the previous iteration for both (5a) and (5b) ends upwith efficient model updates of both the UL and LL learning problems as follows:
[Equation]
where the mixing matrices, with τ g = β/γ and τ f = α/γ , are defined as
[Equation]
According to (6a) and (6b), it can be readily observed that SLAM is amenable to a fully decentralizedimplementation.
The detailed algorithm description is provided in Algorithm 1 from a local view ofthe model update, where [ W ] ij denotes the ij th entry of matrix W , [ h rg ] i,k is the gradient estimateof ∇ g
i,k ( x ri , y ri,k ) (i.e., h rg,k
=
[[ h rg ] 1 ,k , . .
, [ h rg ] n,k ] T ), and similarly [ h rf ] i is the local gradientestimate of ∇ f i ( x ri , y r
+1 i,k ) (i.e., h rf =
[[ h rf ] 1 , . . .
, [ h rf ] n ] T ).
Initialization: α, β, γ , x 1 i , y
i,k , ∀ i, k , and set λ
1 = ω 1 k = 0 , ∀ k ; 1: for r = 1 , 2 , · · · , T do 2: for i = 1 , 2 , · · · , n in parallel over the network do 3:
Estimate gradient ∇ g
i,k ( x ri , y ri,k ) for each task and ∇ f i ( x ri , y r
+
i,k ) locally 4: y r
{++
+}
i,k = 2 (cid:80) j ∈N i
[ W g ] ij y rj,k − [ W
[Equation]
g ] ij y
[Equation]
j,
[Equation]
−
[Equation]
(cid:0) [ h rg ] i,k − [ h r − 1 g ] i,k (cid:1)
▷ LL models 5: x [Equation] − [Equation]
[Equation]
▷ UL model 6: end for 7: end for
Besides, if there is a consensus requirement at only one level of the optimization problem, then theproblem at the other level becomes one with multiple objective functions.
Our proposed SLAMcan also be applied for solving any of these problems by a minor revision of the generic SLAMformulation.
To be more specific, we provide the following discussion.
A Special Case of DBO (1) (with only consensus in the LL problems).
If there is only a need forconsensus of LL model parameters, then problem (2) reduces to the following DBO problem.
Forexample, in solving multi-agent actor-critic RL problems, the UL optimization problem consists ofimproving the policy for each agent while the LL problem requires all the agents to jointly evaluatethe value function over the whole network.
The DBO problem is then expressed as
[Equation]
The major difference between problem (2) and (8) is that the UL optimization problem includesmultiple objectives over the model parameters x
i , ∀ i ∈ [ n ] .
In this case, the updating rule ofvariable x in (6b) reduces to x r +
= x
r − h rf /α by forgoing the dual update w.r.t.
λ .
The detailedimplementation is summarized in Algorithm 2, where we name this special case of SLAM by
SLAM-L as the LL consensus process is the main feature in this setting.
[-Initialization: α, β, γ , x 1 i , y 
 i,k , ∀ i, k , and set ω 1 k = 0 , ∀ k ; 1: for r = 1 , 2 , · · · , T-]Initialization do do i ( x ri , y r
+
i,k ) locally 4:
[-+
 i,k = 2 (cid:80) j ∈N i [ W g ] ij y rj,k − [ W ′ g ] ij y r − 1 j,k − β − 1 (cid:0) [ h rg ] i,k − [ h r − 1 g ] i,k (cid:1)-][Equation]
LL [Equation]
[Equation]
▷ UL models 6: end for 7: end for
A Special Case of DBO (1) (with only consensus in the UL problem).
The other special isanalogous to the first one with the difference being the absence of the LL consensus process incomparison to (2), which is written as follows:
[Equation]
9b
[Equation]
where there are multiple objectives in the LL optimization problems.
Problem (9) also covers a widerange of applications in machine learning, e.g., multi-task and/or personalized learning, and so on.
Inthis case, the update of variable y k shown in (5a) is changed to y r +1 k
=
y rk − h
rg /β as there is noconsensus constraint involved.
Analogous to the previous case, the implementation of this algorithmis presented in Algorithm 3 and termed as SLAM-U. Algorithm
Initialization: α, β, γ , x 1 i , y
i,k {+, ∀ i, k , and set λ 1 = 0 , ∀ k ; 1: for r = 1 , 2 , · · · , T do 2: for i = 1 , 2 , · · · , n in parallel over the network do 3: Estimate gradient ∇ g i,k ( x ri , y ri,k ) for each task and ∇ f i ( x ri , y r +
+}
{+i,k ) locally 4: y r +
 i,k = y ri,k+} − β − 1 [ h rg ] i,k ▷ LL models 5: x r
+1 i = 2 (cid:80) j ∈N
i
[ W f ] ij x rj − [ W ′ f ] ij x r − 1 j − α − 1 (cid:16) [ h rf ] i − [ h r − 1 f ] i (cid:17)
▷ UL model 6: end for 7: end for
Theoretical Convergence Results
Before showing the theoretical results about the convergence guarantees of SLAM, we first need fivemain classes of assumptions used in showing the descent of some quantifiable function so that SLAMcan reach the ϵ -KKT points of the DBO problems.
More detailed definitions and properties regardingthese assumptions are deferred to the supplement.
3.1 Assumptions
Our theoretical results are based on the following assumptions on the properties of the loss functionsin both the UL and LL optimization problems, which are mainly related to the continuity of theobjective function and stochasticity of the gradient estimates.
[Equation]
[Equation] continuity of both UL and LL objective functions) Assume that functionsf i ( · ) , ∇ f i ( · , ) , ∇ g i,k ( · ) , ∇ 2 g i,k ( · ) , ∀ i , are (block-wise)
Lipschitz continuous with constants
L f, 0 , L f, 1 , L g, 1 , L g, 2 for both x and y k , ∀ k .
A2.
(Connectivity of graph G )
The communication graph G is assumed to be well-connected,i.e., 1 T L = 0 where L =
A T A , and the second-smallest eigenvalue of L is assumed to bestrictly positive, i.e., (cid:101) σ min ( A T A ) > 0 .
A3.
(Quality of the stochastic gradient estimate)
The stochastic estimates of ∇ x i f i ( x i , y ri,k ) ,∇ y
i
g
i,k ( x i , y i,k ) , ∇ 2 y i
y
i g i,k ( x i,k , y i,k ) , ∀ i, k , are unbiased and their variances are boundedby σ 2 f , σ 2 g, 1 , σ 2 g, 2 .
Further assume ∥ E [ h rf | x r , y r +1 k ] − ∇ x f ( x r , y r +1 k ) ∥
≤ b r where b r is aquantifiable parameter in terms of the mini-batch size.
A4.
Assume that the UL objective function f () is lower bounded.
A5.
(Strong convexity of g i,k ( · ) w.r.t.
y )
Function g ( · ) is µ g -strongly convex w.r.t.
y i,k , ∀ i, k .
Note that these assumptions are commonly used in the convergence analysis for bilevel and decentralized optimization algorithms.
Given these assumptions, we are now in a position to provide thefollowing theoretical convergence guarantees.
3.2 Convergence Rates of SLAM
Theorem 1.
(Convergence rate of SLAM to ϵ -KKT points) Suppose that A1-A5 hold.
When step-sizesare chosen as 1 /α ∼ 1 /β ∼ O ( (cid:112) n/T ) , τ f , τ g ≥ ρσ max ( A T A ) , the mini-batch size of h rf is
O (log( T )) , and ρ ≥ O ( n ) , then the iterates { x ri , λ r , y rk , ω r , ∀ k, r } generated by SLAM satisfy T T (cid:88) [Equation]
T
[Equation]
10a
[Equation]
where x = n − 1 1 T x , and T denotes the total number of iterations.
Remark 1 .
It is noted in Theorem 1 that the convergence rate achieved by SLAM to find the ϵapproximate KKT points of (1) (including both the first-order stationarity of the solutions and theviolation of constraints) is on the order of 1 / ( nϵ 2 ) .
Therefore, it follows that a linear speedup w.r.t.
thenumber of learners can be achieved by SLAM for DBO, matching the classical results of distributed
SGD for only single-level general nonconvex problems.
Remark 2 .
In comparison with existing bilevel algorithms, SLAM is a single timescale algorithm asthe learning rates satisfy 1 /α ∼ 1 /β , which is consistent with ALSET [38].
Remark 3 .
The major novelty of obtaining these theoretical results relies on the developed variantof the augmented Lagrangian function and subsequently derived recursion of the successive dualvariables, which quantify well the consensus errors resulting from both UL and LL optimizationprocesses in terms of primal variables.
Note that this is distinct from the existing theoretical analysisof stochastic algorithms [33, 35, 36], even including stochastic primal-dual algorithms [37, 40], andalso the classical deterministic primal-dual algorithms [41] for decentralized optimization.
Corollary 1.
(Convergence rate of SLAM-L to ϵ -KKT points)
Given the conditions shown in Theorem 1 and 1 /α ∼ O (1 / √ T ) , the iterates { x ri , y ri , ω rk , ∀ k, r } generated by SLAM-L satisfy
UL
[Equation]
and
LL: (10b) .
Remark 4 .
Different from Theorem 1, the stationarity of the UL model parameters requires theshrinkage of the gradient size over each individual UL problem as shown in Corollary 1, so there isno speedup on the convergence rate guarantee at UL.
Corollary 2.
(Convergence rate of SLAM-U to ϵ -KKT points)
Given the conditions shown in Theorem 1, the iterates { x ri , λ r , y rk , ∀ k, r } generated by SLAM-U satisfy
UL: (10a) and
LL:
[Equation]
Numerical Results
In this section, we evaluate our proposed algorithm using two MARL environments: 1) the cooperativenavigation task [42], which is built on the OpenAI Gym platform [43]; and 2) the pursuit-evasiongame [44], which is built on the PettingZoo platform [45].
Detailed experimental settings andadditional numerical results are provided in the supplement.
[Figure] [Figure]
Cooperative Navigation Task.
In this game, we consider that the n agents are aiming to jointly reachn different landmarks as soon as possible, where the Erdos Renyi Graph is used.
We assume that eachagent can observe the global state and has 5 possible actions: stay, left, right, up, and down.
Thistask consists of a shared common goal of avoiding collision among the agents while they navigate tothe targeting landmarks.
In the simulations, each agent locally maintains two fully connected neuralnetworks as the actor network (at UL w.r.t.
x i ) and the critic network (at LL w.r.t.
y i ), respectively.
Moreover, each agent shares its critic network with its neighbors to cooperatively estimate the globalvalue function and independently train its actor network to complete its local task.
We compare the performance of our proposed SLAM with application to the DAC setting, named
SLAM-AC, with two benchmark algorithms: DAC [15] and mini-batch DAC (MDAC) [17].
Theoretically, MDAC needs an O ( ϵ − 1 ln ϵ − 1 ) batch size in its inner loop to update critic parameters beforeeach update in policy parameters, which is not practical.
Here, we set a small batch B = 10 in theinner loop for MDAC to achieve fast convergence.
The simulation results on this coordination gameare presented in Figure1, where the performance is averaged over 5 independent Monte Carlo (MC)trials for each algorithm.
Pursuit-Evasion Game.
In the pursuit-evasion game, there are two groups of nodes: pursuers(agents) and evaders.
The agents are connected through a ring graph.
Pursuers could observe theglobal state of the video game.
An evader is considered caught if two pursuers simultaneously arriveat the evader’s location.
As each pursuer should learn to cooperate with other pursuers to catch theevaders, the pursuers share certain similarities with each other since they need to follow similarstrategies to achieve their local tasks: simultaneously catching an evader with other pursuers.
We follow the experimental set up in [18], where all agents partially share their actor networks withneighbors for collaborations in their policy spaces and fully share their critic network to cooperativelylearn the global value function.
In Figure 2, we compare SLAM-AC with two benchmarks, CAC [18]and MDAC [17], with 5 MC trials again.
To ensure a fair comparison, all algorithms use the sameparameter sharing scheme mentioned above.
Note that CAC [18] is a variant of DAC [15] and theonly difference is that CAC can partially share its policy parameters while the policy parameters arenot shared in DAC.
In the experiment, each agent maintains two convolutional neural networks, onefor the actor and one for the critic (Please refer to the supplement for detailed structures).  Concluding Remark
In this paper, we studied a generic form of the DBO problem, which is shown to have three majorvariants that formulate multiple hierarchical machine learning problems.
Targeting these DBOproblems, we proposed SLAM – a simple and elegant algorithm to solve DBO in a fully decentralizedway. Under mild conditions, we establish theoretical results showing that our proposed SLAM isable to find the ϵ -KKT points with a convergence rate of O (1 / ( nϵ 2 )) , which matches the standardconvergence rate achieved by the classical distributed SGD algorithms for solving only single-levelgeneral nonconvex optimization problems. We tested the performance of SLAM numerically on a MARL scenario and found that SLAM outperformed the traditional AC algorithms w.r.t.
convergencespeed and (in some cases) achievable rewards.
Societal impact.
To the best of our knowledge, we do not see any ethical or negative immediatesocietal consequence of this work.
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