{"id":"590b8ce0-2f70-4bf9-ade0-e5e297c6a420","arxiv_id":"2602.21321","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"RIDER/E-RIDER dynamically tracks the device-specific symmetric point during analog in-memory training and matches SGD's O(1/√K) convergence while using fewer calibration pulses than two-stage approaches.","lead":"Analog chips that train AI models drift toward a device-specific 'symmetric point' during each weight update, so calibrating them beforehand is expensive. This paper analyzes that calibration cost and introduces RIDER and E-RIDER, algorithms that track the symmetric point during training instead, cutting pulse count while keeping convergence guarantees.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 3.6 (coordinate-wise lower bound on E|∇f|) is the sole driver of SP tracking in Lemma E.3, but it is unvalidated and fails in noiseless/degenerate-gradient regimes; without it Theorem 3.7's dynamic-tracking claim does not follow.","rationale":"The Reader's weakest-assumption identification is correct: Assumption 3.6 is the point where the proof of Theorem 3.7 is least secure. The convergence metric E_K in (14) explicitly includes ∥Gp(Pk)∥², and the only step that makes this term contract is Lemma E.3, which relies on the coordinate-wise lower bound on expected absolute gradient. This is not a mild technical condition: it excludes deterministic or degenerate-gradient regimes, and the paper provides no empirical or theoretical check that it holds in the reported experiments. The proof is not machine-checked and no code is shipped, so the assumption cannot be verified post hoc. I therefore agree with the Reader's CONDITIONAL verdict. A secondary concern that pulse complexity in Corollary 3.9 may count training iterations rather than physical pulses (each analog update requires O(1/Δwmin) pulses under Assumption 3.4) could affect the magnitude of the claimed pulse-complexity advantage, but the convergence guarantee itself stands or falls on Assumption 3.6, so that is the primary load-bearing issue.","tokens_in":40708,"tokens_out":14019,"duration_ms":131226,"concrete_test":"Run Algorithm 2 on a two-coordinate strongly convex quadratic f(W)=μ||W||²/2 with full-batch gradients (σ=0, so Assumption 3.6 fails) and an asymmetric device response Gp(W)≠0. If Pk does not converge to W⋄ while Wk approaches W*, the absence of the −αC⋆∥Gp(Pk)∥² term is confirmed as the reason tracking fails. Alternatively, in the AIHWKit SoftBounds simulator used in Section 4, compute min_{k,d} E_{ξ_k}[|∇_d f(\\bar W_k; ξ_k)|] over a training run; if this quantity is not bounded below by a positive constant independent of k, Assumption 3.6 is not satisfied in the empirical setting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 3.6 is the load-bearing premise of Theorem 3.7. In Lemma E.3 (Appendix E.4), Assumption 3.6 is used to bound E[⟨Gp(Pk), Pk+1−Pk⟩] by −αC⋆∥Gp(Pk)∥²/2 + αqmax²/(2C⋆)∥∇f(\\bar Wk)∥². This negative −αC⋆∥Gp(Pk)∥² term is the only mechanism in the Lyapunov descent that drives Pk toward the symmetric point W⋄. The paper calls the assumption 'mild' because analog devices exhibit thermal/electrical noise, but Assumptions 3.2 and 3.4 only impose upper bounds on gradient and discretization variance; nothing implies a coordinate-wise lower bound on E|∇f(\\bar Wk;ξk)|. At a stationary point, if the gradient estimator is deterministic (full-batch, or coordinates with zero gradient for every sample), E|∇f_d| = 0, so no C⋆ > 0 exists. The stated magnitude C⋆ ≥ 4√2 σ/μ (qmax/qmin)^{3/2} is also substantial for ill-conditioned objectives (μ≪1) and is not validated in the AIHWKit experiments. Without Assumption 3.6, Lemma E.3 gives no descent for φ(Pk), so the proof of Theorem 3.7 does not establish dynamic SP tracking.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies analog in-memory computing (AIMC) training under update asymmetry caused by device-specific symmetric points (SPs). It provides a convergence and pulse-complexity analysis of the standard zero-shifting (ZS) SP-estimation algorithm, proposes RIDER, a residual-learning algorithm with dynamic SP tracking, and an enhanced variant E-RIDER using chopping and filtering. It proves an O(κ1κ2^5/√K)+Θ(Δw_min) convergence bound for RIDER under Assumption 3.6 and derives a pulse-complexity advantage for dynamic tracking versus two-stage residual learning with ZS. Experiments on the AIHWKit simulator with MNIST and CIFAR-100 show that E-RIDER outperforms TT-v2 and AGAD.","tokens_in":41170,"tokens_out":5450,"duration_ms":47884,"significance":"The paper addresses a real and under-theorized problem: SP calibration cost in analog training. The ZS pulse-complexity characterization (Theorem 2.2) is a useful first step, and the idea of tracking the SP during training is timely. The proofs are detailed, with explicit condition-number dependence, and the appendices contain substantial supplementary material. However, the central convergence result rests on an unvalidated and, at stationary points, false coordinate-wise lower-bound assumption, and the experiments test only the empirically-enhanced E-RIDER, not the RIDER algorithm that is the subject of the theorem. If Assumption 3.6 can be justified or replaced with a weaker, validated condition, the contribution would be significant; in its current form the main dynamic-tracking claim is not established.","major_comments":[{"comment":"Assumption 3.6 is load-bearing and unvalidated. In Lemma E.3 (Eq. (100)) the lower bound E_{ξ_k}[|∇f(Ḡ_k;ξ_k)|_d] ≥ C⋆ is used to produce the negative term −(αC⋆/2)∥G_p(P_k)∥², which is the only descent mechanism for φ(P_k) and hence the only driver of SP tracking in Theorem 3.7. The assumption requires the expected absolute gradient to be bounded below in every coordinate at every iteration; this is not implied by Assumptions 3.2–3.4 and fails at stationary points or for deterministic full-batch gradients, where E|∇f_d|=0. The claimed magnitude C⋆ ≥ 4√2σ/μ (q_max/q_min)^{3/2} is restrictive for ill-conditioned objectives (μ≪1). The paper's justification that analog noise makes it mild (Section 3.1) is an assertion, not a validation; no device data or simulator trace is provided. This weakens the central claim.","section":"Section 3.1, Assumption 3.6; used in Lemma E.3 (Appendix E.4)"},{"comment":"Experiments validate E-RIDER, not the algorithm covered by Theorem 3.7. Section 4 says RIDER is a special case of E-RIDER with p=0, but all experiments use the best-tuned chopper probability p>0 (Appendix F.4 shows p=0 is substantially worse). Theorem 3.7 proves convergence for Algorithm 2 (p=0), not Algorithm 3; no convergence theorem is given for E-RIDER. Thus the experimental results do not directly test the theoretical contribution. Either provide a convergence analysis (or at least a formal argument) for E-RIDER, or run RIDER (p=0) in the same settings so the effect of chopping is separated from the core tracking mechanism.","section":"Section 4 and Appendix F.4"},{"comment":"Figure 4 does not validate Corollary 3.9. Corollary 3.9 compares RIDER with two-stage Residual Learning+ZS, whereas Figure 4 compares E-RIDER with TT-v2+ZS. The training pulse count is computed as epochs×⌈data size/B⌉×BL with BL=5, a free parameter not obtained from hardware; the result may change with BL. Also the theoretical pulse complexity is order-level, whereas the plot uses a specific cost model. Please report measured pulse counts from the simulator or show sensitivity of the comparison to BL and batch size.","section":"Figure 4 (Section 4) and Corollary 3.9"}],"minor_comments":[{"comment":"In the introduction, 'decomposing q+(w) and q−(w) into asymmetric component and an asymmetric component' should read 'symmetric component and an asymmetric component'. In Definition 2.1, 'Positive-definiteness' is a misnomer; the condition q_min >0 is a positivity/boundedness condition.","section":"Introduction, Section 2.1"},{"comment":"The metric E_K contains O(·) terms inside the expectation; using big-O inside an expectation is informal. Please replace with explicit constants or a defined norm.","section":"Equation (14)"},{"comment":"The condition C⋆ ≥ 4√2σ/μ (q_max/q_min)^{3/2} appears in the theorem but not in Assumption 3.6. Clarify whether this is an additional assumption or a consequence of other conditions; it is used in the proof to choose C2.","section":"Theorem 3.7 statement"},{"comment":"The condition 'µ_q < 1/(3µ_q Δw_min)' is dimensionally inconsistent; it should likely be 'µ_q < 1/(3Δw_min)' or similar. Please correct.","section":"Appendix C.4, Theorem C.4"},{"comment":"The caption says 'hatched bars indicate the training cost computed as epochs×⌈data size/B⌉×BL' but BL is not defined in the caption. Define BL and state the batch size B explicitly.","section":"Figure 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper's contribution is timely and the ZS pulse-complexity analysis is valuable. However, Assumption 3.6 is the linchpin of the dynamic-tracking proof and is not validated; the experiments also test a different algorithm (E-RIDER) than the one analyzed (RIDER). These issues are fixable — by adding a validation/weakening of Assumption 3.6 and by aligning the experiments with the theory — but they currently block acceptance. I would also suggest the authors double-check the notation consistency in the appendices (e.g., µ_q vs µ_g) and avoid claiming 'first' characterizations without a careful literature check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The reader's take is about right, and the stress-test note lands. This paper has one solid, genuinely new result: Theorem 2.2, the first explicit pulse-complexity bound for the zero-shifting algorithm, with the inverse-linear dependence on Δw_min that Figure 1b empirically confirms. That alone is worth something. The RIDER idea — tracking the symmetric point during training instead of paying a big calibration bill up front — is a natural and useful direction, and the low-pass-filter view of the moving average in Lemma 3.10 is a nice way to explain why chopping helps.\n\nThe soft spots are real and they are in the theory. Assumption 3.6 is not mild. It demands a coordinate-wise lower bound on E|∇f(·;ξ)| for every coordinate and every iteration, with a constant that scales like σ/μ times a device-condition factor. That fails exactly where the theory needs it most: near stationary points, or whenever the gradient estimator is deterministic in some coordinate. The proof of Lemma E.3 uses it to manufacture the negative descent term that drives P_k toward the SP, so without it Theorem 3.7 does not go through. The paper does not validate this assumption on any device model or simulate a regime where it is violated. That is load-bearing, not cosmetic.\n\nThere is also a theory–experiment gap. The theorem covers Algorithm 2 (RIDER, p=0), but the experiments use E-RIDER with a tuned chopper probability and a residual scale γ that are not analyzed. Figure 4's pulse-cost comparison depends on an assumed pulse length BL=5, which is a free parameter, and the paper ships no code or data. The \"same rate as SGD\" phrasing also overstates things a bit — the constants are κ1κ2^5 and there is a Θ(Δw_min) floor.\n\nNone of this is fatal. The ZS complexity result stands on its own, and the dynamic-tracking idea is worth developing. What the paper needs is an honest treatment of Assumption 3.6: either verify it on the AIHWKit device models, weaken it, or state clearly that Theorem 3.7 is conditional on a noise-regime assumption. It also needs experiments on RIDER itself, not just E-RIDER.\n\nI would send this to peer review. A good referee can separate the citable ZS result from the shakier tracking theorem, and the authors seem competent to address the issues. It is a conditional accept in my view, not a reject.","headline":"The zero-shifting pulse-complexity bound is a real result worth citing, but the dynamic-tracking theorem rests on an unvalidated coordinate-wise gradient lower bound, and the experiments test a different algorithm than the one proved.","tokens_in":41600,"tokens_out":1569,"would_cite":true,"duration_ms":18746,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a dynamic tracker can estimate the analog device symmetric point during training, achieving SGD-level convergence while cutting SP-calibration pulse count from O(δ^-2 + δ^-1/Δw_min) to O(δ^-2).","keywords":["analog in-memory computing","symmetric point estimation","zero-shifting","residual learning","update asymmetry","pulse complexity","dynamic tracking","chopping and filtering"],"falsifier":"Run RIDER with deterministic gradients (σ=0) on a device with nonzero SP: Assumption 3.6 is violated, and the predicted behavior is that Q_k stops approaching W⋄ and the training error floor exceeds Θ(Δw_min). Alternatively, log the coordinate-wise expected absolute gradient during training; if any coordinate falls below C⋆ before P_k reaches W⋄, the descent of φ(P_k) in Lemma E.3 changes sign. A pulse-count measurement would also settle it: for Δw_min=10^-4 and target δ≫Δw_min, RIDER should reach target loss with O(δ^-2) pulses while the two-stage zero-shifting baseline needs an additional O(","tokens_in":40621,"feed_emoji":"🎯","tokens_out":6069,"duration_ms":60825,"temperature":0.7,"pith_summary":"Analog in-memory training suffers from update asymmetry that drags weights toward a device-specific symmetric point (SP). Existing methods assume this SP is known and spend many pulses calibrating it to zero; this paper gives the first bound showing that reaching SP-error δ by zero-shifting costs N = O(δ^-1 Δw_min^-1) pulses, so finer devices make calibration more expensive. The paper then proposes RIDER, which leaves the SP unknown and tracks it during training using a digital moving average of a residual device's state. Under strong convexity and a coordinate-wise lower bound on gradient magnitude, RIDER converges at O(κ1κ2^5/√K) + Θ(Δw_min), matching standard SGD, and needs only O(δ^-2) pulses total. An enhanced chopping/filtering variant, E-RIDER, accelerates tracking and is validated on MNIST and CIFAR-100.","feed_headline":"Cuts analog training's calibration pulse cost to O(δ^-2)","feed_subtitle":"A moving-average tracker estimates the device symmetric point on the fly, matching SGD while saving the costly pre-calibration pulses.","key_machinery":"The central mechanism is the decomposition of each pulse response into a symmetric component F and an asymmetric component G, together with the observation that the residual device's update contains a term −α|∇f|⊙G_p(P_k) that always points toward the SP. A digital moving average amplifies this SP-attraction (Lemma 3.5), and the moving average acts as a low-pass filter from P to Q (Lemma 3.10). E-RIDER adds a random chopper sign c_k so the objective-descent component becomes high-frequency while the SP-drift component remains low-frequency, letting the filter extract the drift faster. Assumption 3.6, a coordinate-wise Rayleigh-type lower bound on expected absolute gradient, supplies the cont","core_discovery":"The paper models each analog pulse as a scaled intended update plus an asymmetric drift toward a device-specific symmetric point W⋄ where the asymmetric response G(W⋄)=0. It proves that the standard zero-shifting calibration requires N = O(δ^-1 Δw_min^-1) pulses to reach SP-error δ, so higher-precision devices pay more calibration pulses. It then introduces RIDER: a residual device P is updated by an analog rule whose |gradient| term always pushes P toward W⋄, while a digital moving average Q_{k+1} = (1−η)Q_k + ηP_{k+1} stays closer to W⋄ and supplies the compensation. Theorem 3.7 shows that with strongly convex objectives and step sizes α = Θ(1/√K), β = Θ(αγμ), η = Θ(αμ), the averaged error","pith_inferences":["Because the contraction relies on Assumption 3.6, the strongest version of the result applies only in regimes with sufficient stochastic gradient noise; a noiseless or near-stationary gradient would leave P without a driving force toward the SP. This suggests the guarantees are about training dynamics, not about SP estimation in isolation.","The last-iterate bound for monotone response functions indicates the 1/Δw_min scaling is intrinsic to alternating-pulse zero-shifting, so the dynamic-tracking gain should persist for linear, exponential, and other monotone device models beyond the averaged analysis.","A natural testable extension is to let the SP drift slowly during training (device aging or temperature effects) and check whether the moving average Q_k tracks it; the fixed-point analysis suggests it would, but the paper's theorems assume a static W⋄.","The chopper probability p introduces a practical trade-off: the ablation shows a small p improves accuracy, but a large p would inject high-frequency noise into the descent; an adaptive p schedule is an obvious follow-up."],"forward_implications":["For high-precision devices, the pulse cost of reaching training accuracy δ drops from O(δ^-2 + δ^-1/Δw_min) to O(δ^-2), removing the inverse-granularity calibration bottleneck.","Analog training with unknown, nonzero symmetric point matches the O(1/√K) SGD rate, with only the unavoidable Θ(Δw_min) granularity floor.","The compensation sequence Q_k lives digitally, so the W device needs no extra analog calibration; the SP is tracked on the fly rather than measured beforehand.","E-RIDER's chopping and filtering separate the objective-descent signal from the SP-drift signal in frequency, accelerating tracking without changing the descent direction.","The Θ(Δw_min) error floor means the limiting precision is device response granularity, not reference offset or calibration error."],"fun_headline_variants":["Track analog drift on the fly, skip costly pre-calibration","Dynamic symmetric-point tracking cuts calibration pulses","On-the-fly symmetric point estimation beats fixed calibration","Analog training: estimate drift, not pre-calibrate","Real-time symmetric point tracking saves calibration cost"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof needs every coordinate of the stochastic gradient to have expected absolute value at least C⋆ ≥ 4√2 σ/μ (q_max/q_min)^(3/2) at every step (Assumption 3.6); if any coordinate's gradient is too small, the negative term that drives the residual device toward the symmetric point disappears and the convergence argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Track analog drift on the fly, skip costly pre-calibration","Dynamic symmetric-point tracking cuts calibration pulses","On-the-fly symmetric point estimation beats fixed calibration","Analog training: estimate drift, not pre-calibrate","Real-time symmetric point tracking saves calibration cost"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1180,"prompt_tokens":768,"completion_tokens":412,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":512,"tokens_out":412,"duration_ms":3966,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:04:08.606526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run RIDER with deterministic gradients (σ=0) on a device with nonzero SP: Assumption 3.6 is violated, and the predicted behavior is that Q_k stops approaching W⋄ and the training error floor exceeds Θ(Δw_min). Alternatively, log the coordinate-wise expected absolute gradient during training; if any coordinate falls below C⋆ before P_k reaches W⋄, the descent of φ(P_k) in Lemma E.3 changes sign. A pulse-count measurement would also settle it: for Δw_min=10^-4 and target δ≫Δw_min, RIDER should reach target loss with O(δ^-2) pulses while the two-stage zero-shifting baseline needs an additional O(","supporting_citations":[],"review_version":1}