Avetik Karagulyan

(Université Paris-Saclay, CNRS, CentraleSupélec, L2S)

Title — ELF: Federated Langevin Algorithms with Primal, Dual and Bidirectional Compression.

Abstract — Federated sampling algorithms have recently gained great popularity in the community of machine learning and statistics. This paper studies variants of such algorithms called Error Feedback Langevin algorithms (ELF). In particular, we analyze the combinations of EF21 and EF21-P with the federated Langevin Monte-Carlo. We propose three algorithms: P-ELF, D-ELF, and B-ELF that use, respectively, primal, dual, and bidirectional compressors. We analyze the proposed methods under Log-Sobolev inequality and provide non-asymptotic convergence guarantees.

Bio
I am a Research Scientist at CNRS/L2S. Previously, I was a PostDoctoral fellow at KAUST in the team of professor Peter Richtárik. I have defended my thesis at Center of Research in Economics and STatistics (CREST), Paris under the supervision of professor Arnak Dalalyan. In 2018, I received my MSc Mathematics, Vision, Learning (MVA) diploma at ENS Paris-Saclay with highest honors (mention “très bien”). I graduated from Yerevan State University’s faculty of Mathematics and Mechanics in 2017 with excellence. My research focuses on the study of different methods of sampling and their connections to optimization.

Transcript

1. Langevin sampling, federated learning and their interconnections L2S, 2025 Avetik Karagulyan CNRS / L2S / Université Paris-Saclay 1 1 Based on a joint paper with P. Richtárik
2. Statistical learning Figure: Made with Remarkable tablet A. Karagulyan 2
3. Sampling A. Karagulyan 3
4. Approximate integration Mathematical formulation: Eπ[g(X)] = Rd g(x)π(x)dx. (1) Classical solutions: • LLN based methods (Importance sampling): ˆ In = 1/n n i g(Xi)π(Xi)/ν(Xi), where Xi iid ∼ ν. • Markov chain based methods (MCMC): Construct a chain Xn s.t. L(Xn) = νn ≈ π, then take ˆ In = 1/n n i g(Xi). A. Karagulyan 4
5. Approximate integration Mathematical formulation: Eπ[g(X)] = Rd g(x)π(x)dx. (2) Classical solutions: • LLN based methods (Importance sampling): ˆ In = 1/n n i g(Xi)π(Xi)/ν(Xi), where Xi iid ∼ ν. • Markov chain based methods (MCMC): Construct a chain Xn s.t. L(Xn) = νn ≈ π, then take ˆ In = 1/n n i g(Xi). A. Karagulyan 5
6. Sampling Sampling is widely used in modern Machine Learning: • Bayesian Neural Networks [IVHW21]; • Diffusion models [CCL+22, CHIS23]; • Computer Vision [LZBG20]; • Theoretical ML and generalization [DDB17, RZGS23]; • Bayesian statistics [Rob07, RC13]; • Non-convex optimization [RRT17, Lam21]. A. Karagulyan 6
7. Formulation • Problem: sample from a given target distribution π defined on Rd with a large value of d. • More precisely, for a given precision level ε, construct a probability distribution µ on Rd which is easy to sample from and KL(µ | π) ≤ ε. • Important particular case: π has a density (w.r.t. the Lebesgue measure) given by π(θ) ∝ exp(−F(θ)), with a “potential” F : Rd → R. A. Karagulyan 7
8. KL and FI We are going to use the following probability measure distances • The Kullback-Leibler divergence defined as: KL(µ | ν) = Rd log µ(x) ν(x) µ(dx), if µ ν +∞, otherwise. (3) • The Fisher information is defined as J(µ | ν) =    Rd ∇ log µ(x) ν(x) 2 µ(dx), if µ ν +∞, otherwise. (4) A. Karagulyan 8
9. Approximate sampling as optimization Let us define by P2 (Rd) the family of square integrable measures (equipped with Wasserstein distance): P2 (Rd) = µ : Rd θ 2µ(dθ) < +∞ . (5) Define the functional F : P2 (Rd) → R+ as F(µ) = KL(µ | π). (6) Therefore, approximate sampling becomes the minimization of F on some class C whose elements are easier to sample from: ˆ µ = arg min µ∈C F(µ). (7) A. Karagulyan 9
10. Langevin sampling A. Karagulyan 10
11. Langevin Diffusion • Vanilla Langevin diffusion: dLLD t = −∇F(LLD t )dt + √ 2dWt. (LD) The solution of this equation is a Markov process having π as an invariant distribution: If LLD 0 ∼ π, then LLD t ∼ π, for ∀t > 0. (8) • When the potential function F is λ-strongly convex, the Markov process is ergodic and its distribution converges to π with linear speed [Bha78]: KL(νLD t , π) ≤ e−λt KL(νLD 0 , π). (9) A. Karagulyan 11
12. Langevin Monte-Carlo Vanilla Langevin diffusion (integrated): LLD γ = LLD 0 − γ 0 ∇F(LLD s )ds + √ 2Wγ (10) ≈ LLD 0 − γ 0 ∇F(LLD 0 )ds + √ 2Wγ (11) = LLD 0 − γ∇F(LLD 0 ) + √ 2Wγ , (12) for a small γ. Langevin Monte-Carlo (LMC) is defined as: xk+1 = xk − γ∇F(xk ) + 2γ ξk+1 ; k = 0, 1, 2, . . . (LMC) (ξk )k∈N are i.i.d standard Gaussians independent from xk. This Markov Chain does not preserve π, meaning Xk ∼ π Xk+1 ∼ π. Using a Metropolis-Hastings correction step [RT96] proved asymptotic convergence of LMC in TV. A. Karagulyan 12
13. Unadjusted Langevin Algorithm [Dal17] bounded the error induced by the discretization. That is they show that the sequence νn → πγ, where the latter is the invariant measure of LMC, with stepsize γ. Then they control the error between πγ and π, by choosing γ to be small. Figure: Made with Remarkable tablet. This then led to a series of work studying LMC in various settings - [DM17], [CB18], [DMM19], [DK19], [CDJB19] etc. A. Karagulyan 13
14. Langevin Monte-Carlo: Theorem Theorem Suppose F is λ-strongly convex and L-smooth λId ∇2F(x) LId. If γ < 1/L, then the following upper bound is satisfied: W2 (νn, π) ≤ (1 − λγ)nW2 (ν0, π) + 1.65κ{γd}1/2 (13) where νn is the law of xn and κ = L/λ is the condition number. A. Karagulyan 14
15. Relaxing strong convexity A. Karagulyan 15
16. PL inequality Suppose that g is L-smooth (∇2f (x) LId) and min x∈Rd g(x) = 0. We say that a function g satisfies the Polyak-Łojasiewicz inequality, if for every x ∈ Rd g(x) ≤ 1 2λ ∇g(x) 2. (PL) • If g is λ-strongly convex, it satisfies PL. • If g satisfies the PL inequality and γ < 1/L, then the Gradient Descent (GD) xk+1 = xk − γ∇g(xk) satisfies g(xk+1) ≤ g(xk) + (−γ + Lγ2/2) ∇g(xk) 2 ≤ (1 − λγ/2)g(xk). A. Karagulyan 16
17. Log-Sobolev inequality The Log-Sobolev inequality (LSI) is the analog of PL inequality for the functional F : P2 (Rd) → R. Definition We say the π satisfies the λ-LSI if for every µ F(µ) = KL(µ | π) ≤ 1 2λ J(µ | π), (14) where J(· | ·) is the Fisher information. • If F is λ-strongly convex, then π satisfies λ-LSI (Bakry, Émery ’85). LSI is stable w.r.t. Lipschitz maps and small perturbations LSI remains true (Holley-Strock theorem). • Vempala and Wibisono proved the convergence of LMC under LSI. A. Karagulyan 17
18. Federated Learning A. Karagulyan 18
19. Federated learning “Federated learning is a machine learning setting where multiple entities (clients) collaborate in solving a machine learning problem, under the coordination of a central server or service provider. Each client’s raw data is stored locally and not exchanged or transferred; instead, focused updates intended for immediate aggregation are used to achieve the learning objective.” - [KMA+21] Nowadays, cross-device FL mechanisms are widely used: • Medical research [CKLT18, BCM+18]; • Distributed systems [XIZ+23]; • Gboard mobile keyboard, Android messages, Apple’s Siri [EPK14, Pic19]. A. Karagulyan 19
20. Federated learning paradigm We consider the case when the potential function is sum-decomposable: F(x) = 1 n n i=1 fi (x). (15) • 1 server, n clients • Each fi is stored on the client i. • The clients in parallel compute local gradients. • Compresses and sends them to the server. • Server aggregates, compresses and sends back the new iterate in parallel. Figure: Federated learning protocol. Source: Wiki A. Karagulyan 20
21. Federated learning paradigm We consider the case when the potential function is sum-decomposable: F(x) = 1 n n i=1 fi (x). (16) See [KMY+16, KMA+21] for details. A. Karagulyan 21
22. Speed comparison A. Karagulyan 22
23. optimization → sampling Let us recall the LMC algorithm: xk+1 = xk − γ∇F(xk ) gradient descent + 2γξk noise . (17) In particular, federated learning algorithms can be used for sampling with an additive noise. • LMC + generic SGD [DK19] • LMC + SVRG [CFM+18] • LMC + Proximal GD [BDMS19] • LMC + Mirror GD [HKRC18] • LMC + FedAvg [PMD23] • LMC + MARINA [SSR22] • LMC + QSGD [VPD+22] • LMC + EF21 + EF21-P [KR23] A. Karagulyan 23
24. Compression Definition (Contractive compressor) A stochastic mapping Q : Rd → Rd is a contractive compression operator with a coefficient α ∈ (0, 1] if for any x ∈ Rd, E Q(x) − x 2 ≤ (1 − α) x 2. We denote it shortly as Q ∈ B(α). • Top-k returns the k coordinates with the largest absolute values of its entry. Example: for x = (−4, 3, 10, −1) we have Qtop-2 (x) = (−4, 0, 10, 0) . • The compressor can be biased. A. Karagulyan 24
25. ELF = Error Feedback + Langevin A. Karagulyan 25
26. ELF = Error Feedback + Langevin A. Karagulyan 26
27. Main theorem Theorem Assume that LSI holds with constant λ > 0 and let xk be the iterates of the B-ELF algorithm. We denote by ρk := D(xk ) for every k ∈ N. If each function fi is Li-smooth, then for a small enough step-size γ the following is true for the KL error of the M-ELF algorithm: KL(ρK | π) ≤ e−λKγΨ + τ(γ, d) λ , where Ψ and τ explicitly depend on the parameters of the problem. Taking γ small enough, we can make the second term small. Then we take large enough K to reduce the first one. A. Karagulyan 27
28. Discussion • We do not assume strong convexity of the potential. Instead we assume Log-Sobolev inequality, which is the analog PL inequality. • To obtain ε KL error D-ELF and P-ELF need ˜ O(d/λ2α2ε) iterations. • To obtain ε KL error B-ELF needs ˜ O(d/λ2α4ε) iterations. • The contraction coefficient for top-1 is α = 1/d. • In practice, the algorithms are significantly faster. 0e+005e+041e+052e+052e+052e+053e+05 Bits 0.76 0.78 0.80 0.82 0.84 0.86 Test Accuracy a9a, =1, Top-10 B-ELF P-ELF D-ELF LMC 0e+00 5e+04 1e+05 2e+05 2e+05 2e+05 3e+05 Bits 0.5 0.6 0.7 0.8 0.9 1.0 Test Accuracy mushrooms, =1, Top-10 B-ELF P-ELF D-ELF LMC A. Karagulyan 28
29. This is the last slide. Thank you! A. Karagulyan 29
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