Sparsified Stochastic Gradient Descent with Error Memory for k-Contractive Gradient Compression
Algorithms AIMech:SparsifiedSgdWithErrorMemoryForKContractiveGradientCompression DRAFT
Aliases: Mem-SGD, Sparsified SGD with Memory, k-Contractive Error-Compensated SGD, Top-k SGD with Error Feedback Memory
An error-compensated stochastic gradient descent optimization algorithm (Mem-SGD) that compresses each step's scaled stochastic gradient plus accumulated local residual memory m_t using a possibly biased k-contraction operator comp_k: R^d -> R^d satisfying E ||x - comp_k(x)||^2 <= (1 - k/d) ||x||^2 (such as deterministic top-k magnitude selection or scaled random-k coordinate sparsification), updating the iterate via x_{t+1} = x_t - g_t with g_t = comp_k(m_t + eta_t * grad f_{i_t}(x_t)) and retaining the untransmitted residual m_{t+1} = m_t + eta_t * grad f_{i_t}(x_t) - g_t so that the perturbed virtual sequence x_tilde_t = x_t - m_t evolves via exact uncompressed stochastic gradient steps and recovers the asymptotic O(G^2 / (mu * T)) convergence rate of uncompressed SGD.
Identity Notes & Concept Scope
Distinct from unbiased stochastic quantization (AIMech:QsgdQuantizedStochasticGradientDescentWithEliasCoding, which requires E[Q(g)] = g and cannot analyze biased deterministic Top-k sparsification), from 1-bit sign compression without contractive residual bounds (AIMech:SignSgdAndMajorityVoteCompressedNonconvexOptimization), from EF21 Markov gradient-difference compression (AIMech:Ef21MarkovCompressorErrorFeedbackForHeterogeneousNonconvexOptimization, which compresses gradient differences grad f_i(x^{t+1}) - g_i^t rather than step-scaled gradient plus error memory), and from low-rank matrix subspace compression (AIMech:PowerSgdLowRankGradientCompressionWithErrorFeedback). No pinned AIO class covers k-contractive sparsified SGD with error memory or its perturbed virtual-iterate convergence guarantee.
Claims & Grounded Evidence
4 Claims
Mechanism
#r379-k-contraction-and-virtual-iterate-invariance
Under Definition 2.1 and Lemma A.1, both deterministic top_k magnitude selection and random k-coordinate sampling satisfy the k-contraction property E ||x - comp_k(x)||^2 <= (1 - k/d) ||x||^2; by accumulating the compression error m_{t+1} = m_t + eta_t * grad f_{i_t}(x_t) - g_t where g_t = comp_k(m_t + eta_t * grad f_{i_t}(x_t)), Algorithm 1 ensures that the virtual sequence x_tilde_t = x_t - m_t evolves via x_tilde_{t+1} = x_tilde_t - eta_t * grad f_{i_t}(x_t) with zero cumulative information loss.
Mechanism
#r379-memory-norm-bound-and-uncompressed-leading-rate
Under L-smoothness, mu-strong convexity, and bounded second moment E ||grad f_i(x_t)||^2 <= G^2, Lemma 3.2 and Lemma A.2 prove that the error memory stays bounded as E ||x_t - x_tilde_t||^2 = E ||m_t||^2 <= 4 * eta_t^2 * ((d^2 - k^2) / k^2) * G^2 (tightened to (2*d/k)*(1 - k/d)*eta_t^2*G^2), yielding Theorem 2.2's convergence rate E[f(x_bar_T)] - f(x*) <= O(G^2 / (mu * T)) + O((d/k)^2 * kappa * G^2 / (mu * T^2)) + O((d/k)^3 * G^2 / (mu * T^3)) whose leading O(1/T) term is independent of the compression ratio k/d.
Limitation
#r379-bounded-gradient-and-delay-shift-limitation
The convergence analysis of Mem-SGD relies on the uniform second-moment gradient bound E ||grad f_i(x_t)||^2 <= G^2 and requires an initial stepsize delay parameter a = Omega((d/k) * kappa) so that eta_t = 8 / (mu * (a + t)) changes slowly enough to contract the error memory norm.
Application
#r379-communication-efficient-sparse-training-application
Applied to communication-efficient stochastic optimization of multiclass logistic regression and regularized least-squares models with 10x to 1000x gradient sparsification (top-1 and top-k).
⚠️ Curated Limitations & Negative Results
- Requires a uniform second-moment bound E ||grad f_i(x_t)||^2 <= G^2 on stochastic gradients to bound the error memory norm ||m_t||^2.
- Stepsize schedule eta_t = 8 / (mu * (a + t)) requires an initial warm-up shift a >= max(2*kappa, 4*d/k) proportional to the compression factor d/k.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:SparseLearning
Sparse Learning
|
related_match | Mem-SGD uses k-contractive sparse gradient operators (Top-k and Rand-k) with local error-feedback memory to train statistical and machine learning models. |
AIO:LinearRegression
Linear Regression
|
related_match | Sparsified SGD with memory is analyzed and evaluated on regularized linear and logistic empirical risk minimization objectives. |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
r379-primary-pdf
|
Sebastian U. Stich, Jean-Baptiste Cordonnier, Martin Jaggi
|
primary paper
full_text
|
ID: https://papers.nips.cc/paper/2018/hash/b440509a0106086a67bc2ea9df0a1dab-Abstract.html
sha256:aaa20e82adf7…
|
r379-arxiv-abs
|
Sebastian U. Stich, Jean-Baptiste Cordonnier, Martin Jaggi
|
primary paper
abstract_only
|
ID: https://arxiv.org/abs/1809.07599
sha256:30c0f4bd4a10…
|
r379-metadata-cr
|
Sebastian U. Stich, Jean-Baptiste Cordonnier, Martin Jaggi
|
primary paper
abstract_only
|
ID: https://papers.nips.cc/paper/2018/hash/b440509a0106086a67bc2ea9df0a1dab-Abstract.html
sha256:2358039de76c…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini
Created: 2026-10-19T07:42:00Z
Updated: 2026-10-19T07:42:00Z
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