{"id":"37aaf65c-353c-41ea-8587-d125d99c8399","arxiv_id":"2507.02468","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For LTI systems, the paper proves that closed-loop training data biases subspace-based predictive controllers (SPC, DeePC, gamma-DDPC), decomposes this into Subspace and Optimism Bias, and shows the Transient Predictor avoids both.","lead":"This paper shows why data-driven predictive controllers fail when the training data comes from a closed-loop system: they mistake noise for future help and, for DeePC and gamma-DDPC, also over-trust an adjustable output correction. It proves exact bias formulas and demonstrates that a single-step predictor bank avoids both biases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's maximum-bias formula uses population β in an expression that is exact only with sample β; as stated it is false for finite N in open loop.","rationale":"The paper's central bias decomposition is algebraically coherent: Lemma 1 follows from the innovation-orthogonality property, and the qualitative separation into Subspace Bias and Optimism Bias is well supported by the double-integrator experiments. I do not agree with the reader's named weakest assumption: E[Ef Zp^T] = 0 holds for transient innovations with the chosen ρ, and H is defined for that ρ, so the lead-in length does not threaten the bias identity. The genuinely load-bearing issue is the sample/population conflation in Lemma 3 and Theorem 2, which the reader also noted in the rationale. The equality in Theorem 2a is exact only if β is the sample subspace error, but the symbol β was introduced as a population quantity in (7). This is a correctness gap in the formal statement, but it is repairable by distinguishing β_samp from β_pop and adding the appropriate limit. The qualitative conclusions survive, so the conditional verdict remains appropriate.","tokens_in":9580,"tokens_out":14051,"duration_ms":169311,"concrete_test":"Recompute both sides of Theorem 2a for open-loop double-integrator data with N=100 and N=10000. Set β_samp = L_YV L_VV^{-1} − H from the sampled LQ factors and β_pop from (7) using the true innovations. Evaluate σmax([β_samp L_VV, L_YE]) and σmax([β_pop L_VV, L_YE]); if the former is not σmax(L_YE) at N=100 but converges to it as N grows, the theorem needs an explicit 'as N→∞' quantifier or a sample-β restatement; if they differ at large N, the formula is incorrect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3 derives bγ(γ) = [β L_VV, L_YE]γ, and the algebra forces β = S − H with S = L_YV L_VV^{-1}, the sample subspace predictor from the LQ decomposition. Theorem 2 then states max_{||γ||=1} ||bγ(γ)|| = σmax([β L_VV, L_YE]) using the population β defined in (7). For finite N in open loop, the sample subspace predictor S does not equal H, so the sample β is nonzero. Consequently the open-loop maximum is σmax([β_sample L_VV, L_YE]), not σmax(L_YE); Theorem 2a is false for any finite dataset and holds only in the limit N→∞, a quantifier the theorem does not state. This does not invalidate the qualitative claim that an optimism term L_YE γ_ϵ persists asymptotically, but it makes the formal bound in the central theorem technically incorrect as written and obscures whether the nonzero bias is an asymptotic or finite-sample statement. The reader's alternative concern about E[Ef Zp^T] = 0 and ρ ≥ system order is less compelling: the transient innovation is orthogonal to the ρ lead-in regressors by construction, and H is defined for the chosen ρ.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the bias of subspace-based data-driven predictive control (SPC, DeePC, and gamma-DDPC) for LTI systems. It introduces two bias notions: Subspace Bias, which arises when training data are gathered in closed loop, and Optimism Bias, which arises from the relaxed output constraint in DeePC/gamma-DDPC. The main theoretical results are Lemma 1 and Theorem 1 for the Subspace Predictor, and Lemma 3 and Theorem 2 for DeePC and gamma-DDPC. The paper also argues that the Transient Predictor, based on single-step predictors, is consistent regardless of how the training data are gathered, and presents double-integrator experiments supporting the bias analysis.","tokens_in":9835,"tokens_out":14015,"duration_ms":146516,"significance":"The paper addresses a practically important question: whether subspace-based DDPC methods are biased when training data are generated in closed loop, and whether an alternative predictor architecture avoids this bias. The explicit decomposition into Subspace Bias and Optimism Bias is a useful conceptual contribution, and the numerical experiments illustrate the predicted effect, including the poor performance of SPC with closed-loop data and the persistence of an optimism term for 2norm-DDPC even with open-loop data. The explicit formulas in terms of data correlations are valuable for diagnosing bias in practice. If the formal statements are corrected, the paper would provide a clear cautionary result for users of DeePC and gamma-DDPC and strengthen the case for single-step-predictor-based Transient Predictive Control. The paper does not provide code, and the experimental section would be stronger with error bars, but the qualitative conclusion is supported by the derivations modulo the issues detailed below.","major_comments":[{"comment":"Lemma 3 (Eq. (11)) and Theorem 2 (parts a and b) mix the population Subspace Bias Predictor beta defined in Eq. (7) with the sample quantity actually used in the proof. The proof of Lemma 3 substitutes S = L_YV L_VV^{-1}, which identifies S as the sample subspace predictor Yf V^T (V V^T)^{-1}, not the population predictor S = E[Yf V^T] E[V V^T]^{-1}. Therefore the beta appearing in Eq. (11) must be beta_sample = Yf V^T (V V^T)^{-1} - H, not beta = H_eps E[Ef Uf^T] M. Since beta_sample is nonzero for finite N even in open loop, Theorem 2a is false as stated: with open-loop training data the maximum of ||b_gamma(gamma)|| over ||gamma||=1 is sigma_max([beta_sample L_VV, L_YE]), not sigma_max(L_YE). The statement becomes correct only asymptotically (N to infinity) or after rephrasing with beta_sample and an explicit limit. This is a load-bearing issue because Theorem 2 is the paper's quantitative statement of the DeePC/gamma-DDPC bias.","section":null},{"comment":"The displayed identity E[L_YV] E[L_VV]^{-1} = E[Yf V^T] E[V V^T]^{-1} = S is not valid as written: expectation does not factor through the LQ factors, and L_YV L_VV^{-1} is the sample subspace predictor, not the population S defined in Section II.C. This identity appears to be the source of the sample-versus-population ambiguity in Lemma 3 and Theorem 2. It should either be removed or replaced with a statement that clearly distinguishes the sample equality L_YV L_VV^{-1} = Yf V^T (V V^T)^{-1} from the population definition of S.","section":null},{"comment":"Equation (14) has a dimension and notation problem: the expression for the estimated Subspace Bias Predictor is written as a vector in R^{q tau} and appears to depend on v, whereas beta from Eq. (7) is a matrix of size q tau x ((q+m)rho + m tau). The notation 'cM' is also undefined; it likely should be the estimated version of M. Please rewrite Equation (14) so that the estimator of beta is a matrix and the role of v is made explicit only when the bias beta v is formed.","section":null}],"minor_comments":[{"comment":"The figures report averages over 100 independent experiments but do not show error bars or confidence intervals; adding them would make the convergence claims in Fig. 2 more convincing.","section":null},{"comment":"The consistency of the Transient Predictor is imported from reference [19]; the paper should state the precise theorem or conditions from [19] that guarantee consistency under closed-loop data, and clarify whether Assumption 1 is sufficient.","section":null},{"comment":"There is a typo in 'Multisptep Predictor matrices'; it should be 'Multistep Predictor matrices'.","section":null},{"comment":"No code or data are provided. For reproducibility, especially for the six-trajectory plots, the authors should include code or a detailed experimental protocol with random seeds and parameter values.","section":null},{"comment":"The proof of Lemma 3 is very compressed; after the sample/population issue is resolved, a short derivation showing exactly which beta is used in Eq. (11) and Eq. (12) would improve readability.","section":null}],"recommendation":"major_revision","confidential_remarks":"The paper is technically interesting and the qualitative conclusions are likely correct, but the central theorem as written contains a sample-versus-population inconsistency that must be fixed. The fix is localized: restate Lemma 3 and Theorem 2 with an explicit sample beta or an asymptotic statement, and correct the misleading identity after Eq. (10). I would not recommend rejection; after these corrections the paper could be suitable for publication. The heavy reliance on [19] for the consistency claim is acceptable, but the dependency should be made explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the useful part. The paper introduces a clean split: Subspace Bias (from closed-loop training data) and Optimism Bias (from the L_YE/γ_ε term in DeePC and γ-DDPC). That distinction is real and practically useful. The bias expressions in Lemma 1 for SPC and Lemma 3 for γ-DDPC/DeePC are the right way to see why closed-loop data breaks subspace predictors, and why TPC, built from single-step predictors, does not suffer. The experiments on the double integrator support the qualitative conclusion: SPC degrades with closed-loop data, DeePC-style methods are optimistic in open loop, and TPC works in both.\n\nThe main soft spot is in Lemma 3/Theorem 2. The LQ decomposition is a sample object, but the paper defines β via population expectations in (7). The proof of Lemma 3 uses S = L_YV L_VV^{-1}, which is the sample subspace predictor, and then writes the bias as [β L_VV, L_YE]γ with the population β. For finite N those don't match. The correct expression is [(\\hat{S}-H)L_VV, L_YE]γ. As a result, Theorem 2a as stated—that the open-loop maximum bias is exactly σmax(L_YE)—is not true for a finite dataset; there is an additional sample term that only disappears asymptotically. The qualitative point survives, but the formal theorem needs to either state limiting results or carry the sample β through. This isn't a fatal objection to the paper's thesis, but it is a load-bearing definitional slip in the central quantitative claim.\n\nThe reader's other worry about E[Ef Zp^T]=0 and ρ ≥ system order is less compelling. The transient innovation is orthogonal to the ρ lead-in regressors by construction, and H is the optimal predictor for the chosen ρ, so no extra assumption is needed there.\n\nMinor: no error bars and no code for the experiments, which makes the comparisons suggestive rather than decisive. The citations to the prior closed-loop subspace identification literature are appropriate; the self-citation to [19] for TPC is reasonable since that is where the consistency result comes from.\n\nRecommendation: The paper deserves peer review. The conceptual contribution is solid and the flaw in Theorem 2 is fixable. A serious referee should push for a consistent sample/population statement and for supporting code/data.","headline":"Useful decomposition of closed-loop bias in subspace DDPC, but Theorem 2's finite-sample statement conflates sample and population quantities and needs revision.","tokens_in":10319,"tokens_out":4946,"would_cite":true,"duration_ms":54004,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B30","93C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For LTI systems, subspace-based data-driven predictive control (SPC, DeePC, γ-DDPC) is biased when training data come from a feedback loop; the paper derives the exact bias and shows a single-step-predictor alternative is consistent.","keywords":["data-driven predictive control","subspace predictive control","DeePC","closed-loop identification","prediction bias","optimism bias","transient predictor","LTI systems"],"falsifier":"For a known LTI system, simulate closed-loop training data under a linear feedback law plus external excitation, estimate the Subspace Predictor $\\hat{S}$ and the true predictor $H$ from the known model, and test whether $\\hat{S}v - Hv$ converges to $\\beta v$ with $\\beta = H_\\epsilon \\mathbb{E}[E_f U_f^T]M$. If the mismatch does not converge to this expression (or converges to zero) as $N \\to \\infty$, the central bias claim fails.","tokens_in":9413,"feed_emoji":"🎛️","tokens_out":9626,"duration_ms":100689,"temperature":0.7,"pith_summary":"This paper argues that the most popular data-driven predictive controllers built from subspace predictors—SPC, DeePC, and γ-DDPC—carry a statistical bias whenever the training data are collected under an existing feedback controller, and that the bias does not vanish as data grow. It traces the bias to a precise mechanism: in closed loop, future noise innovations correlate with current and past inputs, and the subspace predictor mistakes this correlation for a causal effect it can exploit. The paper quantifies the resulting Subspace Bias and, for DeePC and γ-DDPC, an additional Optimism Bias from the slack on the output trajectory, giving worst-case bounds in Theorems 1 and 2. It then shows that Transient Predictive Control, built by chaining single-step predictors, is consistent and immune to both biases, and demonstrates the performance gap on a double integrator. If correct, this gives practitioners a concrete reason to prefer single-step-predictor formulations when data must be collected in closed loop.","feed_headline":"Closed-loop data biases subspace predictive control forever","feed_subtitle":"Feedback-collected data cause a persistent prediction error in DeePC; a single-step predictor chaining avoids it.","key_machinery":"The load-bearing object is the Subspace Bias Predictor $\\beta = H_\\epsilon\\,\\mathbb{E}[E_f U_f^T]M$, where $H_\\epsilon$ maps future innovations to future outputs, $E_f$ is the matrix of training-data innovations, $U_f$ is the future input matrix, and $M$ is the weighting matrix from the inverse of the joint covariance of lead-in data and inputs. It captures the anti-causal misinterpretation: the subspace predictor sees that inputs are correlated with innovations and acts as if $u_f$ could set $\\epsilon_f$, although the direction of causality is the reverse. The companion object is the Optimism Bias matrix $L_{YE}$ from the LQ decomposition of the training data, which lets DeePC and $\\gamma$-DDPC hope that the unmodeled innovation $\\gamma_\\epsilon$ will improve the predicted trajectory. The Transient Predictor avoids $\\beta$ by never forming a joint projection of outputs onto lead-in data and future inputs; it estimates single-step ARX predictors and chains them.","core_discovery":"For an LTI system with stationary noise, the paper's central discovery is that the subspace predictor $S = \\mathbb{E}[Y_f V^T]\\mathbb{E}[V V^T]^{-1}$ differs from the true certainty-equivalent multistep predictor $H = [H_p\\ H_u]$ exactly when the training data come from closed-loop operation. Lemma 1 writes the Subspace Bias as $b_S(v) = \\beta v$ with $\\beta = H_\\epsilon\\,\\mathbb{E}[E_f U_f^T] M$, where $M$ is the block-inverse weighting matrix defined in (16); the bias is nonzero precisely because $\\mathbb{E}[E_f U_f^T] \\neq 0$ under feedback. Theorem 1 bounds its worst-case size by $\\sigma_{\\max}(\\beta_V)$ and gives the expected magnitude as a Frobenius norm of $H_\\epsilon\\,[0\\ \\ \\mathbb{E}[E_f U_f^T]]\\,\\mathbb{E}[V V^T]^{-1/2}$. For DeePC and $\\gamma$-DDPC, Lemma 3 shows the prediction bias decomposes into the same Subspace Bias plus an Optimism Bias carried by $L_{YE}\\gamma_\\epsilon$, and Theorem 2 bounds it by $\\sigma_{\\max}([\\beta L_{VV}\\ \\ L_{YE}])$; even with open-loop data the Optimism Bias survives. Transient Predictive Control, which estimates the multistep predictor by chaining single-step ARX predictors, avoids both effects and is consistent under closed-loop data.","pith_inferences":["A direct corollary the authors do not spell out: any subspace-based predictor that forms a joint least-squares projection of future outputs onto past data and future inputs—not just SPC, DeePC, or γ-DDPC—will inherit the same closed-loop bias, so debiasing requires either subtracting $\\hat{\\beta} v$ or switching to chained single-step predictors.","The bias quantifications suggest a practical diagnostic: compute the sample correlation $\\hat{\\mathbb{E}}[E_f U_f^T]$ from closed-loop training data; a near-zero value indicates the Subspace Bias is negligible, while a large value warns that subspace-based DDPC will be optimistic about what inputs can achieve.","For high-order systems where the lead-in length is shorter than the system order, the single-step predictor bank may not be consistent, so the claimed advantage of Transient Predictive Control could shrink; testing this regime would clarify the practical limits.","The same optimism-bias mechanism may affect robust or stochastic variants of DeePC that keep a slack variable on the output; quantifying it for nonlinear systems would require going beyond the LTI setting."],"forward_implications":["With open-loop training data, SPC is unbiased, but DeePC and γ-DDPC still carry the Optimism Bias, so their predictions are biased even when the subspace model is perfect (Theorem 2a).","With closed-loop training data, the Subspace Bias does not decay as the number of training samples grows; Theorem 1b expresses its worst-case size as $\\sigma_{\\max}(\\beta_V)$ and its expected size as a Frobenius norm.","For DeePC and γ-DDPC, setting $\\gamma_\\epsilon = 0$ (or applying a strong projection regularizer) removes the Optimism Bias, leaving only the Subspace Bias; this matches a common practical heuristic.","Transient Predictive Control is consistent under both open- and closed-loop training data and, in the double-integrator experiments, tracks the reference well in both regimes.","The bias terms are estimable from the training data (Section V-A), so a practitioner can estimate $\\beta$ before deploying the controller."],"supporting_citations":[{"why":"Introduces DeePC, the direct data-enabled predictive control formulation whose bias is analyzed in Theorem 2.","marker":"[1]"},{"why":"Introduces the γ-DDPC formulation and its LQ-decomposition structure, to which Lemma 3's bias decomposition applies.","marker":"[5]"},{"why":"Introduces Subspace Predictive Control, the original method whose closed-loop bias is quantified by Lemma 1 and Theorem 1.","marker":"[7]"},{"why":"Derived an earlier expression for the bias of Subspace Predictor parameters under closed-loop data, the prior result this paper extends and refines.","marker":"[10]"},{"why":"Shows that single-step predictors are not biased by closed-loop data, the insight that motivates Transient Predictive Control.","marker":"[14]"},{"why":"Introduces the Transient Predictor method, the consistent multistep predictor that the paper compares against subspace-based DDPC.","marker":"[19]"},{"why":"Documents the projection regularizer that effectively sets the optimism slack to zero, connecting the Optimism Bias to existing practice.","marker":"[21]"}],"fun_headline_variants":["Closed-loop data biases subspace DDPC; TPC escapes","DeePC bias: closed-loop plus optimism; TPC avoids both","Why subspace DDPC fails on closed-loop data","TPC chaining sidesteps closed-loop and optimism bias"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bias formulas and consistency claims rely on the future innovations being uncorrelated with the lead-in measurements and on the lead-in being long enough (at least the system order) that the finite-memory predictor equals the true optimal one.","fun_headline_variants_meta":{"raw":{"variants":["Closed-loop data biases subspace DDPC; TPC escapes","DeePC bias: closed-loop plus optimism; TPC avoids both","Why subspace DDPC fails on closed-loop data","TPC chaining sidesteps closed-loop and optimism bias"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00018,"raw_usage":{"total_tokens":1349,"prompt_tokens":1035,"completion_tokens":314,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":243}},"tokens_in":651,"tokens_out":314,"duration_ms":4093,"temperature":1.0,"reasoning_tokens":243,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:31:13.410334+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a known LTI system, simulate closed-loop training data under a linear feedback law plus external excitation, estimate the Subspace Predictor $\\hat{S}$ and the true predictor $H$ from the known model, and test whether $\\hat{S}v - Hv$ converges to $\\beta v$ with $\\beta = H_\\epsilon \\mathbb{E}[E_f U_f^T]M$. If the mismatch does not converge to this expression (or converges to zero) as $N \\to \\infty$, the central bias claim fails.","supporting_citations":[{"cited_title":"Data-enabled predictive control: In the shallows of the DeePC,","cited_arxiv_id":null,"evidence_quote":"Introduces DeePC, the direct data-enabled predictive control formulation whose bias is analyzed in Theorem 2."},{"cited_title":"Data-driven predictive control in a stochastic setting: a unified framework,","cited_arxiv_id":null,"evidence_quote":"Introduces the γ-DDPC formulation and its LQ-decomposition structure, to which Lemma 3's bias decomposition applies."},{"cited_title":"SPC: Subspace predictive control,","cited_arxiv_id":null,"evidence_quote":"Introduces Subspace Predictive Control, the original method whose closed-loop bias is quantified by Lemma 1 and Theorem 1."},{"cited_title":"Closed-loop aspects of data-enabled predictive control,","cited_arxiv_id":null,"evidence_quote":"Derived an earlier expression for the bias of Subspace Predictor parameters under closed-loop data, the prior result this paper extends and refines."},{"cited_title":"Subspace identification from closed loop data,","cited_arxiv_id":null,"evidence_quote":"Shows that single-step predictors are not biased by closed-loop data, the insight that motivates Transient Predictive Control."},{"cited_title":"The Transient Predictor,","cited_arxiv_id":null,"evidence_quote":"Introduces the Transient Predictor method, the consistent multistep predictor that the paper compares against subspace-based DDPC."},{"cited_title":"Bridging direct & indirect data-driven control formulations via regularizations and relaxations,","cited_arxiv_id":null,"evidence_quote":"Documents the projection regularizer that effectively sets the optimism slack to zero, connecting the Optimism Bias to existing practice."}],"review_version":1}