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Convex Optimization for Big Data
Volkan Cevher, Stephen Becker, Mark Schmidt
TL;DR
Big Data optimization problems can be too large for local processing, making basic linear-algebra routines prohibitive. The article describes composite formulations, approximation techniques, randomization, and parallel computation, finding substantial scalability benefits from these complementary concepts.
Problem
Optimization problems can be too large to process locally, while basic linear-algebra routines such as Cholesky decompositions and matrix multiplications become prohibitive.
Method
The article describes Big Data optimization fundamentals through a composite formulation and scalable approximation strategies including randomized first-order methods and parallel computation.
Results
The three concepts complement each other to provide surprising scalability benefits, including high-probability good solutions from inspecting a negligibly small fraction of the data and linear speed-ups with many processors.
Takeaways & Limitations
Randomized first-order methods can accelerate optimization by producing high-quality solutions with limited data inspection, while parallel computation offers linear speed-ups with many processors.
Abstract
from arXiv · showhide
This article reviews recent advances in convex optimization algorithms for Big Data, which aim to reduce the computational, storage, and communications bottlenecks. We provide an overview of this emerging field, describe contemporary approximation techniques like first-order methods and randomization for scalability, and survey the important role of parallel and distributed computation. The new Big Data algorithms are based on surprisingly simple principles and attain staggering accelerations even on classical problems.
Convex optimization in the wake of Big Data
Big Data makes locally processed optimization and standard linear-algebra routines prohibitive, motivating scalable convex methods that tolerate approximation and moderate accuracy. First-order, randomized, and parallel or distributed techniques provide simple computational primitives, broad scalability, and strong performance on classical problems.
- Convex optimization in the wake of Big Data: Big Data optimization addresses problems whose data and parameter sizes exceed local processing capacity, making basic matrix operations and decompositions prohibitive.The article highlights Cholesky decompositions and matrix-matrix or matrix-vector multiplications as examples.
- First-order methods: First-order methods obtain low- or medium-accuracy solutions from first-order oracle information, including gradient estimates, and can handle non-smooth objectives through proximal mappings.They offer nearly dimension-independent convergence rates and robustness to approximate oracles.
- Randomization: Randomization improves scalability through partial variable updates, statistical gradient or proximal estimators, and randomized linear-algebra routines with controllable expected behavior.These approximations can inspect only a small fraction of the data while producing good-quality solutions with high probability.
- Parallel and distributed computation: Parallel and distributed computation extends first-order methods from synchronous centralized algorithms to asynchronous algorithms with decentralized communications.The framework supports increasing levels of scalability across these computational settings.
- Scalability benefits: Randomized first-order methods can achieve significant acceleration over deterministic counterparts and near-linear speed-ups with many processors.The text attributes the latter to inherently approximate computational primitives and contrasts it with the difficulty of obtaining linear speed-ups using exact methods.
- Classical problems: For LASSO, first-order methods exploit implicit linear operators and nearly dimension-independent convergence, while interior-point methods require more space and near-cubic dimension dependence.The LASSO structure can also enhance first-order convergence, making these methods competitive in accuracy with interior-point methods.
1 First-Order Methods for Smooth and Non-Smooth Convex Optimization
First-order methods provide scalable approaches for smooth and non-smooth convex optimization, with convergence guarantees and practical enhancements that can substantially improve empirical performance. Their efficiency depends on exploiting structure such as smoothness, strong convexity, sparsity, and proximal mappings.
- Core methods: First-order methods use only local function or gradient information and extend to composite objectives through proximal-gradient frameworks.These methods can handle a differentiable convex function together with a non-smooth convex function, and proximal mappings also incorporate constraints.
- Smooth objectives: O(1/ε) iterations are sufficient for gradient descent to reach an ε-accurate solution for convex functions with Lipschitz-continuous gradients.The bound is stated in the worst case and depends on the initial distance to an optimum and the smoothness assumptions.
- Smooth objectives: Nesterov’s accelerated method achieves the best possible worst-case error rate under the Lipschitz-gradient assumptions and is therefore an optimal first-order method.Acceleration adds a momentum step while retaining the fixed step-size choice α_k = 1/L.
- Smooth objectives: Strong convexity yields geometric convergence for gradient descent and enables near-optimal accelerated rates with an appropriate momentum parameter.Adding squared ℓ2 regularization can transform a convex problem into a strongly convex one, as in ridge estimation.
- Practical enhancements: Practical step-size rules and adaptive momentum restarts often improve empirical convergence despite not rigorously improving worst-case rates.Figure 2 shows substantial gains from enhancements, while low per-iteration gradient costs can offset larger iteration counts.
- Composite objectives: Matching the algorithm to the objective’s smoothness structure can outperform enhanced methods based on an incorrect Lipschitz-gradient assumption.For heteroskedastic LASSO, self-concordant structure enables optimal step sizes, while accelerated momentum incurs costlier adaptation.
2 Big Data scaling via randomization
At very large dimensions, exact first-order iterations can become infeasible, so randomized approximations reduce primitive costs while retaining useful convergence behavior. The section surveys randomized coordinate, stochastic-gradient, and low-rank methods for scalable convex optimization.
- Randomized approximations: First-order methods remain robust to approximate gradients and proximal calculations, extending their reach to extraordinary scales.The section focuses on smooth, strongly convex objectives and notes extensions where possible.
- Coordinate descent methods: Random coordinate selection costs independently of p yet achieves the same expected convergence rate as the analyzed greedy choice.Its variance around expected performance is well-controlled.
- Coordinate descent methods: Importance sampling coordinates proportional to their Lipschitz constants improves speed with only an O(log(p)) sampling cost.The resulting rate depends on the mean Lipschitz constant rather than the maximum.
- Stochastic gradient methods: Random data-point selection yields better convergence rates than cycling through data points and produces unbiased gradient estimates.Sampling from the underlying distribution directly targets expected-risk minimization and supports provable generalization capabilities.
- Stochastic gradient methods: Constant-step stochastic gradient methods quickly reduce initial error despite retaining non-vanishing optimization error.Large step sizes and weighted averaging can achieve optimal convergence rates while remaining robust to step-size and modeling assumptions.
3 The role of parallel and distributed computation
Parallel and distributed computation can reduce the cost of Big Data optimization, but communication and synchronization constrain practical speedups. The section presents centralized, decentralized, coordinate-based, consensus, and asynchronous strategies for addressing those constraints.
- Motivation: Big Data’s computational and storage demands, together with plateauing transistor efficiencies and power costs, increase reliance on parallel and distributed computation.The article frames this shift as necessary for handling resources at reasonable power costs.
- Hardware constraints: Distributed first-order methods face communication and synchronization bottlenecks, especially when links are uneven or workers have unequal runtimes.Global coordination can slow computation when one machine takes much longer than others.
- Decomposition methods: Consensus ADMM provides an extremely scalable framework by decomposing the problem across local copies that converge to a common solution.The decomposition approach supports massively parallel optimization for n > 2.
- Embarrassingly parallel methods: For decomposable objectives, gradient computation can be distributed with O(n/m) local computation and data samples per machine.Each processor handles corresponding data locally before contributing to the aggregate computation.
- Decentralized methods: Decentralized gradient updates among neighboring machines achieve convergence rates similar to centralized communication, with degradation determined by the network graph Laplacian.This removes the need for a central communication location with only minor algorithmic modifications.
- Asynchronous methods: Asynchronous, lock-free updates can preserve stochastic-gradient convergence under certain conditions and provide substantial speedups when many cores are available.The same memory model also applies to stochastic parallel coordinate descent and can tolerate communication failures in randomized first-order methods.
4 Outlook for convex optimization
The outlook calls for structure-dependent approximation trade-offs, greater use of composite models, and algorithms adapted to heterogeneous computational platforms. These directions aim to scale convex optimization while addressing noise, constraints, synchronization, and communication realities.
- Outlook: Big Data problems require a fundamental overhaul of convex optimization algorithm design.The article links this need to increasingly large problems and modest growth in computational resources.
- Future directions: Future approximation tools should adapt convex algorithms to heterogeneous computational platforms shaped by synchronization and communication constraints.The article also anticipates increased use of composite models and proximal-mapping principles for noisy, constrained nonsmooth problems.
- Composite models: Composite models can provide stronger estimation guarantees than least squares under sparsity and matrix assumptions.The example is the LASSO formulation when the signal has at most k non-zero entries and Φ obeys specified assumptions.
- Approximation trade-offs: The article emphasizes identifying structure-dependent algorithmic approximation trade-offs for larger convex optimization problems.It presents this as necessary for controlling computational-resource growth.