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Federated stochastic bilevel optimization with fully first-order gradients
Yihan Zhang, Rohit Dhaipule, Chiu C Tan, Haibin Ling, Hongchang Gao
TL;DR
Federated stochastic bilevel optimization is computationally expensive because existing methods require second-order Jacobian and Hessian computations. The paper introduces a fully first-order variance-reduced federated algorithm with a constant single-timescale learning rate and a new convergence-analysis strategy. Author-reported experiments confirm efficacy and greater computational efficiency than existing second-order methods.
Problem
Existing federated stochastic bilevel optimization algorithms require costly second-order Jacobian and Hessian computations, while practical first-order federated learning also needs suitable constant and single-timescale learning rates.
Method
The paper develops a federated stochastic variance-reduced bilevel gradient descent algorithm using fully first-order gradients, constant single-timescale learning rates, and a potential-function convergence analysis.
Results
Experiments confirm the algorithm’s efficacy and show greater computational efficiency than existing second-order methods.
Takeaways & Limitations
The proposed approach reduces reliance on second-order computations while providing convergence guarantees for first-order federated bilevel optimization.
Takeaways & Limitations
The analysis assumes a nonconvex upper-level loss and a lower-level loss that is strongly convex in its second variable.
Abstract
from arXiv · showhide
Federated stochastic bilevel optimization has been actively studied in recent years due to its widespread applications in machine learning. However, most existing federated stochastic bilevel optimization algorithms require the computation of second-order Hessian and Jacobian matrices, which leads to longer running times in practice. To address these challenges, we propose a novel federated stochastic variance-reduced bilevel gradient descent algorithm that relies solely on first-order oracles. Specifically, our approach does not require the computation of second-order Hessian and Jacobian matrices, significantly reducing running time. Furthermore, we introduce a novel learning rate mechanism, i.e., a constant single-timescale learning rate, to coordinate the update of different variables. We also present a new strategy to establish the convergence rate of our algorithm. Finally, the extensive experimental results confirm the efficacy of our proposed algorithm.
1 Introduction
The paper studies federated stochastic bilevel optimization under computational, learning-rate, and convergence challenges. It proposes a fully first-order variance-reduced method with a constant single-timescale learning rate and convergence guarantees.
- Existing federated stochastic bilevel methods require second-order Jacobian and Hessian matrices, creating high computational costs for high-dimensional problems.
- Federated first-order methods also need constant learning rates to avoid revealing the current training stage and single-timescale rates to simplify practical tuning.
- The paper develops a federated stochastic variance-reduced bilevel gradient descent algorithm using fully first-order gradients and a single-timescale constant learning rate.
- The method establishes convergence guarantees by combining different estimation errors with carefully designed coefficients in a potential function.
- Experiments confirm that the proposed algorithm is more computationally efficient than existing second-order methods.
2 Related Works
Related work shows that federated stochastic bilevel methods commonly estimate hypergradients with second-order information. Fully first-order methods avoid this cost in single-machine settings, but first-order federated bilevel algorithms were not previously available.
- Single-machine fully first-order methods reformulate the lower-level problem with a penalty approach, but the penalty hyperparameter significantly affects convergence.
- Existing federated bilevel algorithms use Neumann-series or related Hessian-inverse-vector estimators, but still require second-order Jacobian and Hessian computations.
- Before this work, no first-order federated bilevel optimization algorithms were known.
3 Algorithm Design
The algorithm reformulates bilevel optimization through a penalty-based minimax problem and solves it with federated stochastic variance-reduced first-order updates. Constant learning rates coordinate the variables while periodic aggregation supports federated training.
- 3.1 Problem Definition: The penalty approach converts the constrained bilevel problem into an unconstrained minimax problem whose penalized solution approximates the original objective for an appropriate penalty parameter.
- 3.2 Our Algorithm: FedSVRBGD-FO updates upper- and lower-level variables using stochastic variance-reduced gradients that do not require second-order Jacobian or Hessian matrices.
- 3.2 Our Algorithm: Devices periodically upload local variables and gradient estimators to the central server, which resets them to global values.
- 3.2 Our Algorithm: The learning rate η and associated hyperparameters are constant rather than iteration-dependent, with βxη^2 < 1 as a stated condition.
4 Convergence Analysis
Under standard smoothness, strong-convexity, bounded-variance, and heterogeneous-worker assumptions, the analysis establishes convergence for the proposed algorithm. The proof uses a penalty-based approximation, a constant single-timescale learning rate, and a potential function to obtain linear device speedup.
- Assumptions: The analysis assumes upper-level smoothness and Lipschitz conditions, lower-level smoothness and µ-strong convexity in y, bounded stochastic-gradient variance, and heterogeneous worker data.These conditions are stated in Assumptions 4.1–4.5.
- Penalty approximation: The penalized objective approximates the bilevel objective with |L*λ(x) − L(x)| = O(ℓκ^2/λ) when λ > 2Lf/µ.The condition number is defined as κ = ℓ/µ, with ℓ = max{Cf, Lf, Lg,1, Lg,2}.
- Learning-rate design: Algorithm 1 uses a constant learning rate independent of iteration and applies the same timescale to its three variables, unlike prior time-dependent two-timescale updates.The paper defines the same timescale as having the same order with respect to ϵ or the iteration count.
- Proof strategy: The proof combines estimation-error components in a potential function whose coefficients are selected to eliminate associated terms in the one-step potential difference.Coefficients linked to λ and λ^2 are designed so the coefficient set remains independent of λ, avoiding a smaller learning rate and slower convergence.
- Proof strategy: A consensus-error lemma characterizes how the penalty hyperparameter λ and communication period p affect consensus error before the final convergence bound is derived.This lemma is used as a key step in establishing the convergence rate.
5 Experiments
Experiments on hyperparameter optimization and hyper-representation learning evaluate runtime, accuracy, convergence sensitivity, and communication-period effects. The proposed method generally converges faster than baselines while maintaining comparable accuracy.
- 5.1 Hyperparameter Optimization: The algorithm requires much less time to converge than all baseline methods across the hyperparameter-optimization datasets.Experiments ran all methods for 16,000 upper-level updates with communication period p = 4.
- 5.1 Hyperparameter Optimization: It reaches almost the same test accuracy as baseline methods while using substantially less time on a9a.Figure 2 reports test accuracy versus running time for a9a with communication period p = 4.
- 5.1 Hyperparameter Optimization: Smaller ϵ produces slower convergence because Theorem 4.8 prescribes a smaller learning rate for higher-accuracy solutions.The experiment varies ϵ on a9a with communication period p = 4.
- 5.1 Hyperparameter Optimization: With communication period p = 16, the proposed algorithm still converges faster than all baselines, although FedNest becomes more efficient than FedBiOAcc.The comparison includes upper-level loss and test accuracy versus running time.
- 5.2 Hyper-representation Learning: The hyper-representation experiment evaluates upper-level loss over running time on a deep-learning task with nonconvex upper-level and strongly convex lower-level optimization.The lower level learns classifier weights, while the upper level learns hidden-layer weights.
6 Conclusions
The paper presents a federated stochastic bilevel algorithm using fully first-order gradients, constant single-timescale learning rates, and a new convergence-analysis strategy. Experiments support its efficacy and reduced running time.
- 6 Conclusions: The algorithm uses first-order stochastic variance-reduced gradients for both variable updates, avoiding second-order Hessian and Jacobian computations.This design is reported to save running time.
- 6 Conclusions: A single-timescale constant learning rate coordinates the updates, while a novel convergence strategy establishes the algorithm’s rate.The paper’s conclusion links these designs to the reported experimental efficacy.
A.1 Fundamental Lemmas
The fundamental lemmas establish smoothness and strong-convexity properties for the auxiliary lower-level updates, together with bounds on consensus and tracking errors. These ingredients support the later convergence proof.
- A.1 Fundamental Lemmas: The lemmas bound optimization, tracking, and consensus-error terms under stepsize restrictions involving α_x, α_y, α_z, and η.These inequalities are repeatedly combined in the convergence analysis.
- A.1 Fundamental Lemmas: The auxiliary y-subproblem is strongly convex when λ > 2L_f/μ and is L_y-smooth with L_y = L_f + λL_g,1.The corresponding stepsize is selected using α_y ≤ 1/(6L_y).
- A.1 Fundamental Lemmas: The z-subproblem is λμ-strongly convex and λL_g,1-smooth, enabling a stepsize choice α_z ≤ 1/(6λL_g,1).The subproblem minimizes λg(x,z) with respect to z.
A.2 Proof of Theorem 4.8
The proof of Theorem 4.8 combines several error components in a specially weighted potential function and selects parameters to control the resulting inequalities. These choices yield the stated accuracy guarantee.
- A.2 Proof of Theorem 4.8: The convergence proof applies the fundamental lemmas and sequentially enforces parameter inequalities to eliminate error terms from the potential-function bound.The proof uses bounds involving η, λ, p, batch size, and variance parameters.
- A.2 Proof of Theorem 4.8: The proof introduces a novel potential function that combines upper-level loss, lower-level tracking, auxiliary tracking, consensus, and gradient-estimation errors.Carefully chosen coefficients weight the component terms.
- A.2 Proof of Theorem 4.8: For ϵ ∈ (0, 1), the theorem selects hyperparameters and initialization conditions to obtain an ϵ-accuracy solution.The stated initialization conditions include O(ϵ^2) bounds for lower-level and auxiliary errors.