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REVIEW 3 major objections 5 minor 26 references

Gradient Descent Methods for Regularized Optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A variable-step proximal gradient descent that estimates the local Lipschitz constant at each iteration outperforms fixed-step proximal GD and Adam on ℓ1-regularized least squares.

desk verdict Mostly a clean review; the new adaptive proximal GD is promising but lacks a convergence proof and can take step sizes outside its own theory. read the letter →

arxiv 2412.20115 v1 pith:WHUPACTK submitted 2024-12-28 math.OC cs.LG

Filip Nikolovski , Irena Stojkovska , Katerina Hadzi-Velkova Saneva , Zoran Hadzi-Velkov This is my paper · ORCID
classification math.OCcs.LG MSC 90C2590C0665K05
keywords proximalgradientdescentvariablestepsizelocalLipschitzconstantℓ1regularizationLASSOsoft-thresholdingconvexoptimizationregularizedleastsquares
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a proximal gradient descent variant for ℓ1-regularized least squares that replaces the fixed step size $1/L$ (where $L$ is the global Lipschitz constant of the smooth part) with an adaptive step size estimated from the local geometry at each iteration. The step size is the reciprocal of a local smoothness estimate, obtained by comparing successive gradients and iterates, and it is adjusted with a safeguard that decreases the step when it comes too close to the unsafe ratio and increases it slowly otherwise. The paper argues that because smoothness is a local property, local estimates should perform better than a global constant, and the numerical experiments on three synthetic datasets and one real real-estate dataset support that claim: the variable-step method needs fewer iterations, less total time, and reaches lower objective values than both fixed-step proximal GD and the Adam optimizer. The paper also provides a self-contained overview of GD and proximal GD convergence results ($O(1/\varepsilon)$ for convex and $O(\log(1/\varepsilon))$ for strongly convex objectives) and derives the soft-thresholding proximal operator for the ℓ1 norm. No convergence theorem is given for the new variable-step rule; its boundedness is asserted by analogy to an earlier method.

What carries the argument

The load-bearing object is the norm-ratio step-size rule in Algorithm 2. After each proximal step it compares $\|\nabla f(x_{k+1})-\nabla f(x_k)\|$ with $(\mu_0/\lambda_k)\|x_{k+1}-x_k\|$; if the gradient change is too large relative to the current step, the next step is set to $\lambda_{k+1}=\mu_1\|x_{k+1}-x_k\|/\|\nabla f(x_{k+1})-\nabla f(x_k)\|$, and otherwise $\lambda_{k+1}=\lambda_k+\min\{\lambda_k,1\}\eta_k$ with $\sum_k\eta_k<\infty$. This keeps the step size below the reciprocal of the local Lipschitz estimate while preventing it from collapsing to zero. The companion object is the proximal operator for the ℓ1 regularizer, the soft-thresholding map $\operatorname{prox}_{\lambda\alpha\|\cdot\|_1}(z)=\operatorname{sgn}(z)\max\{|z|-\alpha\lambda,0\}$, applied component-wise; it is what makes each proximal step cheap enough that the added ratio computation is worthwhile.

What would settle it

Run Algorithm 2 on a sequence of quadratic problems with known curvature and compare the generated $\lambda_k$ with the true local Lipschitz constant: if $\lambda_k$ exceeds $1/L_k$ while the objective is still far from optimal, or if $\lambda_k$ approaches zero without the objective approaching its minimum, the boundedness premise fails and the method's practical gains are not backed by a mechanism. A simpler experiment: on the same synthetic problems, if a fixed step exactly $1/L$ reaches the same objective value in less wall-clock time than the adaptive rule whenever the adaptive step is prevented from exceeding $1/L$, then the improvement is an artifact of the stopping criterion rather than of local adaptation.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the practical performance of proximal gradient descent for ℓ1-regularized problems can be improved by estimating the local Lipschitz constant of the smooth component at each iteration and using its reciprocal as the step size. Concretely, after computing $x_{k+1}=\operatorname{prox}_{\lambda_k}(x_k-\lambda_k\nabla f(x_k))$, the method forms the ratio $\|\nabla f(x_{k+1})-\nabla f(x_k)\|/\|x_{k+1}-x_k\|$ as an estimate of the local smoothness constant $L_k$; if the current $\lambda_k$ is larger than a safe fraction $\mu_0$ of the reciprocal ratio, the step is shrunk to $\mu_1$ times that ratio, and otherwise it is increased by a small decaying amount. In the reported experiments this adaptive schedule eliminates the need to compute the global constant $L$ (the largest eigenvalue of $\frac{1}{2m}A^TA$), reduces iteration counts by more than half on the synthetic problems, and beats constant-step proximal GD and Adam on both synthetic data and the King County house-price dataset, despite a slower per-iteration speed. The step sizes produced by the rule move both above and below $1/L$, which the paper reads as evidence that the method is tracking local geometry rather than fitting the global bound.

Load-bearing premise

The load-bearing premise is that the adaptive step sizes stay bounded away from zero and never exceed the safe local bound, so the iterates keep converging; the paper asserts this 'similarly as in [11]' but does not prove it, and the empirically tuned constants $\lambda_0$, $\mu_0$, and $\mu_1$ are part of that unverified mechanism.

Editorial extensions

If this is right

  • Users of proximal GD for LASSO can drop the expensive computation of the global Lipschitz constant $L$ and still obtain faster convergence in practice on large sparse-regression problems.
  • The variable-step rule preserves the fixed-step method's complexity order ($O(\log(1/\varepsilon))$ for $L$-smooth, $\mu$-strongly convex objectives) while using fewer iterations and less wall-clock time in the reported tests.
  • The method beats Adam on the tested ℓ1-regularized problems, suggesting that for sparse recovery a proximal update that respects nonsmoothness can outperform a momentum-based adaptive-gradient update.
  • The step sizes move around $1/L$ during runs, so the method is tracking local curvature rather than simply using a larger constant step, which explains why it helps on ill-conditioned stretches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the boundedness of $\{\lambda_k\}$ can be proven, the same norm-ratio schedule would give a parameter-light adaptive proximal method for stochastic or incremental settings where a global Lipschitz constant is unknown or changes over time.
  • The empirical tuning of $\lambda_0$, $\mu_0$, and $\mu_1$ suggests a testable question: whether a fully self-tuning version, with $\mu_0$ and $\mu_1$ adjusted by the observed gradient ratios, would retain the speedups without hand-picked constants.
  • The comparison with Adam is limited to full-batch ℓ1 least squares; a natural extension is to check whether the same gap persists for logistic loss, group sparsity, or nonconvex regularizers, where the proximal operator is still cheap.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper surveys gradient descent and proximal gradient descent for composite convex optimization, proves standard O(1/n) and linear convergence rates under L-smoothness and strong convexity, and derives the soft-thresholding proximal map for L1 regularization. Its proposed contribution is Algorithm 2, a proximal gradient method with a variable step size that estimates the local Lipschitz constant at each iteration, together with numerical experiments on three synthetic datasets and the King County real estate dataset comparing the method with constant-step proximal gradient descent and Adam. The survey portions are largely correct, but the central novel claims—that Algorithm 2 is convergent and that it reliably outperforms constant-step proximal gradient methods—are not backed by rigorous theory and are supported by experiments with important methodological weaknesses.

Significance. If the variable-step proximal gradient method were rigorously shown to converge and to retain the O(log(1/epsilon)) complexity claimed in Section 4, it would be a practically useful alternative in settings where the global Lipschitz constant is expensive or unavailable. The paper's standard convergence theorems for constant-step proximal gradient are correct and clearly presented, and the paper correctly emphasizes that smoothness is a local property. However, the proposed algorithm's step-size rule is not covered by the paper's own convergence theory, and the empirical evidence is not sufficient to establish the performance claim. The central contribution therefore stands or falls on additional theoretical and numerical work that the current manuscript does not provide.

major comments (3)
  1. [Section 4, Algorithm 2, Eqs. (33)-(38); Theorems 3 and 4] The iterates generated by Algorithm 2 are not covered by the paper's convergence theorems. Since f is L-smooth, the norm ratio R_k = ||x_{k+1}-x_k|| / ||grad f(x_{k+1}) - grad f(x_k)|| satisfies R_k >= 1/L, but it has no finite upper bound. In the true branch of Algorithm 2 (line 6), lambda_{k+1} = mu_1 R_k can therefore exceed 1/L even though mu_1 < 1. In the false branch (line 8), the condition only enforces lambda_k <= mu_0 R_k, so lambda_k may already be larger than 1/L, and the update increases lambda_k further. Lemma 1 and Theorems 3 and 4 all require 0 < lambda <= 1/L, so the step sizes generated by Algorithm 2 can leave the theoretically valid regime. The statement in Section 4 that {lambda_k} is bounded 'similarly as in [11]' is not a proof, and even boundedness would not imply lambda_k <= 1/L. Section 6 explicitly defers theoretical results to future work, so the convergence of Algorithm 2 is currently unproven. Figure 3 even confirms that the selected step sizes are sometimes greater than 1/L. The authors should either prove that Algorithm 2 keeps lambda_k in a range where convergence is guaranteed, or modify the algorithm and prove a new convergence theorem for larger step sizes.
  2. [Section 5, Table 1 and Figures 1-4] The numerical evidence does not support the claim of 'notable performance improvement' as stated. The parameters lambda_0 = 0.1, mu_0 = 0.99, and mu_1 = 0.95 are empirically chosen on the same test problems ('we have empirically established that appropriate values for the parameters are...'), so the comparison partially reflects tuning rather than a fixed, prespecified method. The experiments use only three synthetic instances and one real dataset, report average CPU times from 7 runs without variance or error bars, and provide no statistical significance assessment. The stopping rule is also asymmetric: the proximal methods stop when F(x_{k+1}) > F(x_k), while Adam is exempted from this criterion because it is 'non-monotone by design.' Since Algorithm 2 can produce step sizes larger than 1/L, this non-monotonicity stop may terminate at a point where the function value has only temporarily increased, which can favor the variable-step method; no evidence is given that the stopping point is near a true minimizer. The empirical claim would need substantially stronger validation, including fixed parameters chosen by a separate procedure, multiple random trials, error bars, and a fair stopping rule.
  3. [Section 5, synthetic data generation and Eq. (39)] The synthetic 'optimal solution' x* is not a minimizer of the objective in Eq. (39). The authors generate b = A x* + xi with xi having i.i.d. standard normal entries, so grad f(x*) = A^T xi / m is generally nonzero, and the optimality condition for the composite objective (39), 0 in grad f(x*) + alpha partial ||x*||_1, is not satisfied for the stated construction. Therefore Figure 2, which plots ||x_k - x*|| as a distance to the 'optimal solution,' does not measure distance to a true optimum of the problem being solved, and the objective-value comparisons in Figure 1 are not benchmarked against the correct optimal value F(x*). This undermines the synthetic validation of the central performance claim. The experiments need to be redesigned so that the true minimizer of (39) is known or computed independently.
minor comments (5)
  1. [Section 2.2 and Introduction] The phrase 'exponential (i.e., superlinear) convergence' is inaccurate for the rate in Theorem 2: the bound decays linearly in the multiplicative factor (1 - mu/L), i.e., exponentially in the iteration count, which is normally called linear convergence, not superlinear.
  2. [Section 4, Algorithm 2] The input specification lists '0 < mu_1 < mu_0 < 1' but never states the domain or smoothness assumptions on f and g for Algorithm 2 beyond the earlier global L-smoothness assumption; the paper should state explicitly which of the theorems in Section 3 are assumed to apply to the variable-step setting.
  3. [Section 5, Table 1] The table header contains a typo, 'Algorthm' instead of 'Algorithm', and the column headers repeat the spelling; the table also reports only average times without standard deviations, which would be useful even in a preliminary comparison.
  4. [Section 5, stopping criteria] The stopping rule based on ||grad f(x_k)|| < 0.001 is not equivalent to a stationarity condition for the composite objective F in Eq. (39), because grad f(x_k) alone does not account for the subgradient of the L1 term; the stopping criterion should be described in terms of a residual of the proximal-gradient fixed-point condition.
  5. [Section 6, conclusion] The sentence 'Algorithms 1 and 2 have similar complexities of order O(log(1/epsilon)) for the case of L-smooth, mu-strongly convex functions' is not established anywhere for Algorithm 2; since no convergence theorem exists for the variable-step method, this claim should be phrased as an empirical observation or removed until a proof is provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Algorithm 2 is an openly heuristic variable-step variant; the main weaknesses are a missing convergence proof and test-set-tuned constants, neither of which reduces the paper's claims to their inputs.

full rationale

The convergence results (Theorems 1-4, Lemmas 1-2) are stated for Algorithm 1 with constant step size and are proved in the text using standard external sources; none of these proofs is replaced by a self-citation. The proposed Algorithm 2 is explicitly presented as a heuristic: its step-size rule is 'based on (33)' and 'inspired by similar approaches outlined in [11] and [12]', and the paper does not claim a first-principles derivation of convergence for the variable-step rule. The sentence 'it can be shown that the sequence {λ_k} is bounded from above and below away from zero, similarly as in [11]' is an unproved assertion, and Section 6 explicitly defers theory ('Providing theoretical results for these insights would definitely be worth exploring in depth in the future'); this is a correctness/support gap, not circular reasoning. The numerical comparison uses constants 'empirically established' on the same test problems (λ0=0.1, μ0=0.99, μ1=0.95), and the experiments still compute L for comparison; this weakens the reported improvement but does not make any predicted quantity equal to a fitted input by construction. The only author self-citation ([4], Dimovski and Stojkovska) is background on MASSO and is not load-bearing. No equation is shown to be equivalent to its own input, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The theoretical results rely on standard convexity and smoothness assumptions. The algorithm introduces four free parameters, with lambda_0, mu_0, and mu_1 tuned empirically on the test problems and eta_k unspecified. No new physical or mathematical entities are postulated.

free parameters (4)
  • lambda_0 = 0.1
    Initial step size, set empirically.
  • mu_0 = 0.99
    Upper threshold for step size shrinkage, tuned on test datasets.
  • mu_1 = 0.95
    Shrinkage factor applied when step size is too large, tuned on test datasets.
  • eta_k = not specified
    Summable positive sequence for controlled step size increase; concrete choice never given.
assumptions (4)
  • domain assumption f is continuously differentiable, convex and L-smooth with Lipschitz constant L.
    Used throughout Section 2 and the proximal theorems.
  • domain assumption f is mu-strongly convex for the accelerated convergence results (Theorems 2 and 4).
    Invoked in Sections 2.2 and 3.2.
  • domain assumption g is continuous, convex, possibly non-smooth.
    Standard assumption for proximal GD in Section 3.
  • ad hoc to paper The local Lipschitz constant at iteration k can be estimated by the ratio ||grad f(x_{k+1}) - grad f(x_k)|| / ||x_{k+1} - x_k||.
    This is the basis of the step size rule (33) in Section 4; it is assumed without proof to yield a valid local smoothness bound.

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Cite this review

Pith. "Pith review of Gradient Descent Methods for Regularized Optimization." pith.science (2026). https://pith.science/paper/WHUPACTK

@misc{pith2026241220115,
  author       = {Pith},
  title        = {Pith review of: Gradient Descent Methods for Regularized Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WHUPACTK}},
  note         = {Machine review of arXiv:2412.20115}
}
abstract

Regularization is a widely recognized technique in mathematical optimization. It can be used to smooth out objective functions, refine the feasible solution set, or prevent overfitting in machine learning models. Due to its simplicity and robustness, the gradient descent (GD) method is one of the primary methods used for numerical optimization of differentiable objective functions. However, GD is not well-suited for solving $\ell^1$ regularized optimization problems since these problems are non-differentiable at zero, causing iteration updates to oscillate or fail to converge. Instead, a more effective version of GD, called the proximal gradient descent employs a technique known as soft-thresholding to shrink the iteration updates toward zero, thus enabling sparsity in the solution. Motivated by the widespread applications of proximal GD in sparse and low-rank recovery across various engineering disciplines, we provide an overview of the GD and proximal GD methods for solving regularized optimization problems. Furthermore, this paper proposes a novel algorithm for the proximal GD method that incorporates a variable step size. Unlike conventional proximal GD, which uses a fixed step size based on the global Lipschitz constant, our method estimates the Lipschitz constant locally at each iteration and uses its reciprocal as the step size. This eliminates the need for a global Lipschitz constant, which can be impractical to compute. Numerical experiments we performed on synthetic and real-data sets show notable performance improvement of the proposed method compared to the conventional proximal GD with constant step size, both in terms of number of iterations and in time requirements.

Figures

Figures reproduced from arXiv: 2412.20115 by the authors.

Figure 1
Figure 1. Comparison of the progress of the sequence [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the progress of the sequence [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Changes of the step sizes λk generated by the scheme given in Algorithm 2 on the synthetically generated data sets for different values of d. The value of the constant step λ = 1/L is given for comparison. For all three dimensions, 1/L ≈ 0.64. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison of the progress of the sequence [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

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