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Incremental Stochastic Subgradient Algorithms for Convex Optimization
S Sundhar Ram, A Nedich, V. V. Veeravalli
TL;DR
The paper addresses stochastic errors in decentralized constrained optimization when each agent knows only its own convex objective component. It analyzes cyclic and Markov-randomized incremental subgradient methods, using stochastic-analysis arguments to establish almost-sure convergence with diminishing step-sizes and error or performance bounds with constant step-sizes.
Problem
The paper studies how stochastic subgradient-evaluation errors affect constrained incremental optimization with agent-specific convex objective functions.
Method
It analyzes cyclic ring updates and Markov-randomized updates, with the latter governed by a time non-homogeneous Markov chain, using iterate relations and supermartingale techniques.
Results
With diminishing step-sizes, the cyclic method's iterates converge almost surely to an optimal solution under the stated assumptions, while constant step-sizes yield expected-value and almost-sure performance bounds.
Takeaways & Limitations
The analysis provides convergence guarantees for diminishing step-sizes and performance guarantees rather than iterate convergence for constant step-sizes.
Takeaways & Limitations
The results are asymptotic, and the paper does not provide expected convergence-rate bounds or an exhaustive application analysis.
Abstract
from arXiv · showhide
In this paper we study the effect of stochastic errors on two constrained incremental sub-gradient algorithms. We view the incremental sub-gradient algorithms as decentralized network optimization algorithms as applied to minimize a sum of functions, when each component function is known only to a particular agent of a distributed network. We first study the standard cyclic incremental sub-gradient algorithm in which the agents form a ring structure and pass the iterate in a cycle. We consider the method with stochastic errors in the sub-gradient evaluations and provide sufficient conditions on the moments of the stochastic errors that guarantee almost sure convergence when a diminishing step-size is used. We also obtain almost sure bounds on the algorithm's performance when a constant step-size is used. We then consider \ram{the} Markov randomized incremental subgradient method, which is a non-cyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time non-homogeneous Markov chain. Such a model is appropriate for mobile networks, as the network topology changes across time in these networks. We establish the convergence results and error bounds for the Markov randomized method in the presence of stochastic errors for diminishing and constant step-sizes, respectively.
1. Introduction.
The paper frames randomized incremental optimization for networks with arbitrary structure and distinguishes distributed objective data from distributed decision-vector components.
- Randomized incremental methods allow networks with arbitrary structure.
- Unlike settings with distributed decision-vector components, this paper distributes objective-function data across agents.
2. Problem Formulation and Motivation.
The paper formulates constrained minimization of a sum of agent-specific convex functions and studies cyclic and Markov-randomized incremental methods under stochastic errors.
- Problem Formulation and Motivation.: A network of m agents minimizes a sum of convex functions over a closed convex set, with each fi known only to agent i.
- Problem Formulation and Motivation.: In the cyclic method, agents pass an iterate around a directed ring and each updates it using a subgradient of its own objective.
- Problem Formulation and Motivation.: The cyclic update projects each noisy subgradient step onto X, using a positive step-size and random error term.
- Problem Formulation and Motivation.: The arbitrary-connectivity method selects the updating agent according to a distribution conditional on the previous updater.
- Problem Formulation and Motivation.: The Markov randomized method models the updating-agent sequence as a time non-homogeneous Markov chain with transition probabilities [P(k)]i,j.
- Problem Formulation and Motivation.: With zero errors and uniform transition probabilities, the Markov randomized method coincides with the randomized incremental method studied previously.
- Problem Formulation and Motivation.: The paper analyzes both algorithms for diminishing and constant step-sizes, including zero- and non-zero-mean errors.
- Motivation: Stochastic errors can represent computational round-off or approximate subgradients formed from sequential random samples when agents lack distributional statistics.
Distributed Regression.
The distributed regression application uses sensor measurements to select a parameterized spatial-field model, with least-squares structure yielding convex local objectives in the linear case.
- Distributed regression selects the parameter x whose candidate model best describes a spatial field from sensor measurements.
- With i.i.d. measurement noise, the least-squares relation is transformed into an expected-loss formulation.
- In linear least-squares regression, each local objective fi(x) is convex in x.
Distributed Resource Allocation.
The paper also places its framework in wireless fair-rate allocation, where constrained flow rates maximize locally known rewards under network-capacity and delay restrictions.
- Fair rate allocation adjusts the rates of m network flows over a directed-link wireless network.
- Flow-rate vectors must satisfy link-capacity and possibly queuing-delay constraints, defining the feasible set X.
- Each flow has a reward function depending on its rate and known only at its source node.
- Reward functions may also depend on traffic content when message types have different importance across time slots.
- The traffic-type statistics may be unknown because they depend on external factors such as intruder frequency.
- Notation and Basics.: The paper uses Euclidean norms, inner products, vector components, matrix entries, rows, columns, and the all-ones vector in its notation.
- Notation and Basics.: The optimal value and optimal set are denoted by f* and X*, respectively.
- Notation and Basics.: The analysis relies on the subgradient inequality for convex functions.
3. Cyclic Incremental Subgradient Algorithm.
The cyclic incremental stochastic subgradient method analyzes projected, noisy subgradient updates under moment assumptions on the errors. Diminishing step-sizes yield almost sure convergence, while constant step-sizes yield performance bounds rather than iterate convergence.
- Algorithm: The cyclic method updates intermediate iterates with projected subgradients of each agent’s local objective, corrupted by random errors and scaled by αk+1.The update passes through all agents in sequence, with xk carried from one cycle to the next.
- Analysis: The central technical difficulty is that the expected update direction need not be a subgradient of the summed objective.The proof therefore uses a problem-specific iterate relation together with supermartingale convergence arguments.
- Assumptions: The analysis assumes closed convex feasibility, convex component functions, and conditionally bounded first and second moments of the subgradient errors.Independent errors with finite moments provide one example satisfying the stated error assumptions.
- Diminishing step-size: With diminishing step-sizes and a nonempty optimal set, the iterate sequence converges to an optimal solution with probability 1.The convergence argument establishes almost sure convergence of the distance sequence and objective values before concluding convergence to X∗.
- Constant step-size: With a constant step-size, iterate convergence is not guaranteed, but expected objective values and the best function value admit performance bounds.The best-function-value bound depends on α and the error-moment bounds; under zero-mean errors, µ=0, and the bound can be controlled through α.
4. Markov Randomized Incremental Subgradient Method.
The Markov randomized incremental method selects updating agents through a time non-homogeneous Markov chain, enabling incremental optimization over arbitrarily connected and changing networks. Under connectivity, persistence, and stochastic-error assumptions, diminishing step-sizes yield almost sure convergence, while constant step-sizes yield performance bounds.
- Method: Each update is performed by one agent, which processes its local objective and passes the iterate to a neighbor or processes it again according to transition probabilities.The updating sequence is modeled as a time non-homogeneous Markov chain.
- Method: The method supports arbitrary network connectivity and time-varying neighbor sets, making it suitable for mobile or otherwise changing networks.Agents communicate only with neighbors, whose relationships may change over time.
- Analysis: The analysis addresses dependence between the selected agent and the current iterate by using convergence-rate estimates for Markov-chain transition products.The uniform steady state makes agents update their objective functions with equal steady-state frequency.
- Assumptions: Strong connectivity, positive self-transition probabilities, uniformly positive transition weights, and double stochasticity ensure persistent information and a uniform limiting distribution.These assumptions give each agent a persistent opportunity to update and equal long-run update frequency.
- Results: With diminishing step-sizes, the method achieves almost sure convergence in function value and approaches the optimal set, while constant step-sizes produce error bounds rather than iterate convergence.The diminishing-step-size results include lim inf f(xk) = f* and lim inf dist(xk, X*) = 0 with probability 1.
- Results: With uniform transition probabilities and no errors, the constant-step-size bound is better by a factor of m than the corresponding bound in the prior randomized incremental method.For this case, choosing T = 0 is optimal and yields the stated simplified bound.
5. Discussion.
The discussion frames incremental algorithms as semi-cooperative network optimization methods that can achieve a system-level global optimum. The results are asymptotic, while expected convergence-rate bounds and mobile-agent alignment applications remain possible extensions.
- 5. Discussion.: Incremental algorithms balance selfish local cost updates with cooperation through passing iterates among neighboring agents.Each agent adjusts the iterate using its own cost function, then passes it onward so another agent can contribute.
- 5. Discussion.: Theorems 3.3 and 4.3 show that this semi-cooperative structure can still obtain a system-level global optimum.
- 5. Discussion.: The paper’s convergence results are asymptotic rather than finite-time guarantees.
- 5. Discussion.: Combining the basic iterate relation with techniques from could provide bounds on expected convergence rates.
- 5. Discussion.: The Markov stochastic sub-gradient results may help design alignment algorithms for coordinating mobile agents.