{"id":"0e0342b6-3f7c-42b0-9b46-20dfdd336614","arxiv_id":"2507.22632","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The sample complexity of MMD and adversarial domain-adaptive networks is upper-bounded by O(d^2 L^2 / eps^2), and the target-loss weight should scale as O(sqrt(M_t)).","lead":"This paper derives mathematical limits on how much data semi-supervised domain adaptation models need, for two common neural-network designs. It finds that the needed data grows quadratically with network depth and width, and recommends shrinking the weight on target labels as the square root of the number of such labels.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 1 (Eq. 3) makes loss differences Lipschitz in a label-free feature-mean distance; it rules out label shift and is effectively the result being claimed.","rationale":"The concern is load-bearing because it targets the single assumption connecting unlabeled alignment to label-dependent loss. The reader identified Assumption 1/12; I agree. The paper is otherwise internally coherent: the covering-number calculations in Lemmas 7-8 and the derivations in Theorems 3-4 are consistent given the assumptions. The concrete label-shift construction shows the assumption is not merely 'cannot be verified' but is actually violated in a standard DA scenario, so the central claim does not extend to that scenario. I would keep the conditional verdict: the theoretical results are valid as conditional statements, but the paper should either prove Assumption 1 for a nontrivial model class, replace it with a jointly testable condition, or explicitly state the assumption's restrictiveness. The experimental section is weak (fitted curves, no error bars, some extrapolated points) and Section 5.1 contains an inserted passage ('Figure 5.7', 'DASGA', etc.) that appears copied from another source and should be removed or explained; these support but do not drive the conditional verdict.","tokens_in":59634,"tokens_out":12009,"duration_ms":143347,"concrete_test":"Run the label-flip experiment: source X~U([-1,1]^2), labels y_s=sign(x_1); target X same marginal, labels y_t=-sign(x_1). Set f_s=f_t=identity (or one learnable layer) and fix h(x)=x. Compute the empirical MMD \\hat D and the losses L_s,L_t. If \\hat D≈0 while |L_s-L_t|≈1, then Eq. (3) fails for finite R, so Theorem 2's bound (1-α)R\\hat D cannot control the target loss. Equivalently, an analytic test: derive the minimal R for this construction and show it must be infinite. Also verify whether the function classes F_s,F_t used in the experiments contain constant maps; if so, the same violation occurs within the stated class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 1 (Eq. 3) asserts a finite R such that |L_s(f_s,h)-L_t(f_t,h)| ≤ R D(f_s,f_t) for all f_s∈F_s, f_t∈F_t, h∈H. In the MMD setting, D(f_s,f_t)=||E[f_s(x_s)]-E[f_t(x_t)]||_X (Eq. 7) is a first-moment distance and carries no label information. Thus any label shift with matched feature means violates the inequality: take f_s=f_t=constant (included in typical parameter-bounded nets, e.g. zero weights), source with Y=sign(x_1), target with Y=-sign(x_1), and h(x)=x. Then D=0 but |L_s-L_t|=1 for any bounded activation, so no finite R exists. Assumption 12 (Eq. 37) has the same structure, with D_Δ a scalar mean, and additionally depends on which Δ the min-max algorithm actually returns; the paper does not prove the learned Δ certifies Eq. 37. Since Lemma 1 and Theorem 1 convert Assumption 1 into the chain L_t ≤ L_α+(1-α)R D, Theorems 2-4 inherit it, and the claimed O(d^2L^2/ε^2) sample complexity applies only to problems where the hardest (label-shift) part of DA is already assumed away. The manuscript offers no derivation of R, no class of distributions provably satisfying Eq. 3, and no empirical check of the assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theoretical framework for semi-supervised domain adaptation in which source and target feature transformations and a shared classifier are learned jointly by minimizing a weighted empirical classification loss together with a distribution discrepancy. Section 2 states a general bound (Theorem 1) on the expected target loss in terms of the empirical weighted loss and the true discrepancy, under an assumed Lipschitz-type relation between the loss gap and the discrepancy (Assumption 1). Theorem 2 replaces the true MMD by its empirical estimate using covering-number and concentration arguments. Section 3 specializes these results to MMD-based and adversarial deep networks, deriving covering-number bounds for the relevant function classes (Lemmas 7-8, Corollary 1) and sample-complexity statements (Theorems 3-4) of the form O((d^2 L log(L/epsilon) + d^2 L^2 log d)/epsilon^2), with the target-loss weight recommended to scale as alpha = O(sqrt(M_t)). Section 5 reports synthetic and image experiments whose fitted curves follow the predicted O(L^2), O(d^2), and O(sqrt(M_t)) trends.","tokens_in":59949,"tokens_out":8152,"duration_ms":100537,"significance":"If the two relatedness assumptions were justified, this would be a substantial contribution: it would give one of the first explicit width/depth sample-complexity bounds for alignment-based domain-adaptive neural networks, and it would provide a concrete, testable prescription for the target-loss weight. The modular proof strategy is a real strength: Lemma 2 combines Hoeffding bounds with covering numbers, Lemma 4 uses Yurinskii's inequality for uniform MMD deviation, and the covering-number analysis for multi-layer kernel feature maps in Lemmas 7-8 and Corollary 1 is nontrivial and carefully worked out. The experimental section is also commendable for attempting to verify the scaling laws in both shallow and deep settings. However, the headline claims are conditional on Assumption 1 (Eq. 3) and Assumption 12 (Eq. 37), and those assumptions are neither derived from data properties nor verified empirically; for label-shift settings they can fail while all measurable premises of the theorems hold. The contribution is therefore best read as a conditional analysis of alignment-based DA rather than as an established sample-complexity theory.","major_comments":[{"comment":"Assumption 1 is not a mild relatedness condition; it is the load-bearing mechanism of the entire paper. With D chosen as the first-moment MMD in Eq. (7), take f_s and f_t to be zero-weight constant maps (admissible under the parameter bounds of Assumption 5), source labels Y = sign(x_1), target labels Y = -sign(x_1), and h(x) = x. Then D(f_s, f_t) = 0 while |L_s(f_s,h) - L_t(f_t,h)| = 1, so no finite R exists. This shows that the assumption excludes label shift, which is one of the standard domain-adaptation challenges named in the Introduction. Since Lemma 1 and Theorems 1-4 all use inequality (3) directly, the claimed O(d^2 L^2/epsilon^2) sample complexity holds only for problems in which the label-shift part of the transfer has already been assumed away.","section":"Section 2.2, Eq. (3)"},{"comment":"Assumption 12 has the same structural problem as Assumption 1 and an additional algorithmic dependence. The scalar distance D_Delta(f_s,f_t) is a mean difference of the discriminator outputs, so it carries only first-moment information about the discriminator's view of the features. Moreover, the inequality (37) is assumed only for the discriminator Delta returned by the min-max optimization in Eq. (31). Lemma 9 provides uniform concentration for sup_{f_s,f_t,Delta} |D_Delta - hat-D_Delta| over all Delta in D, but it does not prove that the particular Delta selected by the algorithm certifies inequality (37). The proof of Theorem 4 in Appendix M invokes Assumption 12 at the first line without any derivation of this certification property. The adversarial result is therefore subject to both the label-shift concern and an unverified algorithmic condition.","section":"Section 3.2, Eq. (37)"},{"comment":"The stated sample-complexity rates quote only O(d^2 L^2/epsilon^2) and hide the constants R and R_A. Since the target-loss bound contains the terms (1-alpha) R hat-D and (1-alpha) R epsilon (and the adversarial analogue), the value of R multiplies both the empirical discrepancy and the accuracy parameter. If R grows with d or L, as is possible when the loss gap and the first-moment discrepancy are not tightly coupled, the R-dependent constants could dominate the quoted scaling. The paper should either prove an explicit upper bound on R and R_A in terms of the data and network constants, or state explicitly that all scaling statements are modulo these assumption constants.","section":"Theorems 3 and 4, Eqs. (29) and (39)"},{"comment":"The experiments do not provide evidence for Assumptions 1 and 12. Figures 5, 7-13 fit the predicted rates O(1/sqrt(M_t)), O(L^2), and O(d^2) to target accuracy or required sample sizes, but they do not measure the discrepancies D or D_Delta, do not estimate R or R_A, and do not test inequality (3) or (37). A meaningful empirical check of the conditional claim would require either a distribution class for which R is known to be finite, or a diagnostic that directly compares the loss gap with the discrepancy on held-out data. Without such a check, the experiments support only the concentration-rate structure of the bounds, not the relatedness assumptions on which the main theorems rest.","section":"Section 5"}],"minor_comments":[{"comment":"The text calls epsilon 'a probability constant'; in Eq. (29) epsilon is the accuracy parameter that controls the generalization gap. This terminology should be corrected to avoid confusion.","section":"Section 3, paragraph after Theorem 3"},{"comment":"Remark 5 states that the alpha = O(sqrt(M_t)) choice handles target-label scarcity 'independently of the complexity of the target function class', but the displayed alpha bound in Theorems 3 and 4 contains d^2 L log(L/epsilon) + d^2 L^2 log d in the denominator. The remark should be reworded to match the formula.","section":"Section 4, Remark 5"},{"comment":"The experimental plots show fitted dashed curves but no confidence intervals or repetitions over multiple random seeds; reporting variability would make the support for the claimed scaling laws more convincing.","section":"Section 5.1, Figures 5, 7"},{"comment":"The manuscript contains LaTeX-editor artifacts such as 'Font size: 28 in latexit' in the figure captions and internal labels 'Figure 5.7/5.8' that should be cleaned before publication.","section":"Figures 2 and 3 and Section 5.1"}],"recommendation":"major_revision","confidential_remarks":"I considered recommending rejection because Assumptions 1 and 12 effectively assume away label shift, which is a central difficulty in domain adaptation. However, the covering-number analysis and the conditional bounds are potentially salvageable, and a major revision that substantially weakens or verifies the relatedness assumptions, tracks R and R_A explicitly, and adds direct empirical checks could make the contribution publishable. If the authors instead retain the current assumptions without justification, the headline sample-complexity claim should be reframed as conditional on a very strong and untested condition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: the covering-number analysis is real, and the α = O(√M_t) rule is a clean takeaway. But the paper's headline sample-complexity result only holds if one accepts Assumption 1, and that assumption is doing far more work than the paper admits. It essentially says the loss gap is bounded by a feature-mean distance, which fails for straightforward label shift. So the practical scope is narrower than the title suggests.\n\nWhat's genuine: For MMD-based and adversarial networks, the paper gets explicit O(d^2 L^2 / ε^2) bounds on Ms, Ns, Nt, with a corresponding α = O(√M_t) target-loss weight. The covering-number estimates for the function classes induced by these architectures are the real contribution; the appendices are thorough and standard. I believe those derivations are correct, and the derivation of the α rule from the exponential term is not circular. They also clearly distinguish their setting from same-space DA and transfer learning.\n\nWhere it's soft: Assumption 1 (Eq. 3) and its adversarial twin Assumption 12 (Eq. 37) are assumed, not derived. The stress-test example is legitimate: constant embeddings give D=0 while the loss gap can be nonzero under label shift, so no finite R exists without additional constraints. The paper doesn't give a class of distributions where the assumption holds, nor any empirical check. For adversarial, Assumption 12 additionally depends on the learned discriminator being strong enough, but they don't prove the min-max game yields that. The experiments are supportive but they fit curves to data without error bars, so they read as illustrative rather than confirmatory. And the stray inserted passage in Section 5.1 (Figure 5.7, DASGA, etc.) is clearly copied from elsewhere and has to go.\n\nBottom line: This is a serious theoretical paper with detailed proofs, and it deserves a careful referee. But the referee should push hard on the assumptions. If Assumption 1 cannot be weakened or verified for some non-trivial family, the contribution is mostly a conditional analysis. I'd like to see them characterize when Assumption 1 holds, or at least give an example of a real distribution family that satisfies it. As it stands, the claim is really 'sample complexity under a strong relatedness condition.' My recommendation: send it to review, but expect major revisions on the assumption handling.","headline":"Honest covering-number analysis with a useful α = O(√M_t) rule, but the load-bearing relatedness assumption can fail on simple label-shift and is never justified.","tokens_in":60530,"tokens_out":3836,"would_cite":true,"duration_ms":47077,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q32","68T07","68T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a sample-complexity law for domain-adaptive neural networks: for both MMD-based and adversarial models of width $d$ and depth $L$, the labeled-source and total source and target samples needed to hold the target…","keywords":["domain adaptation","semi-supervised learning","generalization bounds","sample complexity","maximum mean discrepancy","adversarial domain adaptation","covering numbers","deep neural networks"],"falsifier":"Take a digit dataset and a target domain with the same input distribution but permuted class labels, so the marginal features can be aligned to near-zero MMD or discriminator confusion while the classification-loss gap stays large, and measure the ratio $|L^s - L^t|/D(f^s, f^t)$ across hypotheses; if that ratio can be driven arbitrarily large while the alignment loss is small, Assumption 1 or 12 fails and the constant $R$ in the bounds degenerates. Conversely, on benchmarks where the ratio stays bounded, the predicted quadratic growth of required sample sizes with depth and width, and the $\\alpha \\propto \\sqrt{M_t}$ schedule, are directly checkable by fitting the minimum $M_s$ and $N_s$ needed to hold a fixed target accuracy.","tokens_in":59356,"feed_emoji":"🧠","tokens_out":14567,"duration_ms":152148,"temperature":0.7,"pith_summary":"This paper sets out a unified theory of generalization for domain alignment algorithms: methods that map source and target data into a shared feature space through learned transformations and train a single classifier there, in the semi-supervised regime where target labels are scarce. Its central quantitative claim is a scaling law for deep domain-adaptive networks: whether alignment is enforced by maximum mean discrepancy or by an adversarial discriminator, the number of labeled source samples and the total number of source and target samples needed to keep the target generalization gap at $O(\\epsilon)$ is $O((d^2 L \\log(L/\\epsilon) + d^2 L^2 \\log d)/\\epsilon^2)$, i.e., quadratic in the network width $d$ and depth $L$ up to log factors. It further claims that the weight $\\alpha$ of the target classification loss should shrink as $\\alpha = O(\\sqrt{M_t})$ with the number of labeled target samples, turning label scarcity into an automatic downweighting of the target term. A reader would care because this converts a heuristic design choice, namely how large a domain-adaptive network can be trained from a given label budget, into a testable quantitative law, and it is the first explicit sample-complexity characterization for this family of models.","feed_headline":"Sample needs grow at most quadratically with network size","feed_subtitle":"MMD and adversarial domain adaptation obey the same depth-width law; scarce target labels call for α ∝ √M_t.","key_machinery":"The engine of the argument is a covering-number analysis of the function classes induced by the network: the feature transformation classes $\\mathcal{F}^s, \\mathcal{F}^t$ that map each domain into the shared space, and the composite hypothesis classes $\\mathcal{H} \\circ \\mathcal{F}^s$, $\\mathcal{H} \\circ \\mathcal{F}^t$ formed by a transformation followed by the classifier, plus, in the adversarial case, the classes $\\mathcal{V}^s = \\mathcal{D} \\circ \\mathcal{F}^s$ and $\\mathcal{V}^t = \\mathcal{D} \\circ \\mathcal{F}^t$ of discriminator-and-feature-extractor compositions; a covering number of a function class is the minimum number of small balls needed to cover it, so it measures how complex the class is. Lemma 7 and Lemma 8 build explicit parameter-grid covers of these classes, yielding the growth rate $O((L/\\epsilon)^{d^2 L} (c d)^{d^2 L^2})$ for width $d$ and depth $L$ (Corollary 1). Lemma 2 then controls the deviation between expected and empirical weighted losses by a union bound over these covers, while Lemma 4 (MMD) and Lemma 9 (adversarial) control the deviation between the true and empirical domain discrepancy via Yurinskii's exponential inequality and Hoeffding's inequality, respectively; Theorem 2 combines the pieces for the MMD case, and Theorems 3 and 4 turn the union-bound probability expression into sample-complexity statements by choosing sample sizes so that each failure term stays bounded.","core_discovery":"The paper's central claim, stated on its own terms, is a sample-complexity law for domain-adaptive neural networks. For an MMD-based network of width $d$ and depth $L$ (Theorem 3), keeping the expected target loss within $O(\\epsilon)$ of the empirical weighted loss plus the empirical alignment term requires the number $M_s$ of labeled source samples and the total numbers $N_s, N_t$ of source and target samples all to be of order $O((d^2 L \\log(L/\\epsilon) + d^2 L^2 \\log d)/\\epsilon^2)$, with the target-loss weight set to $\\alpha = O((M_t \\epsilon^2/(d^2 L \\log(L/\\epsilon) + d^2 L^2 \\log d))^{1/2})$: the $O(\\sqrt{M_t})$ scaling when the network size is held fixed. For adversarial networks (Theorem 4) the same law holds for $M_s$, while the unlabeled totals become $O((d^2 (L+K) \\log((L+K)/\\epsilon) + d^2 (L+K)^2 \\log d)/\\epsilon^2)$, with $L+K$ the combined depth of the feature extractor and the domain discriminator. The paper also claims, at the general level of Theorem 1 and Theorem 2, that for any domain alignment algorithm the expected target loss is bounded by the empirical weighted loss plus $(1-\\alpha)$ times the discrepancy between the aligned domains, with failure probability decaying exponentially in the labeled sample sizes once the covering numbers of the relevant function classes are accounted for.","pith_inferences":["A practical corollary the authors leave implicit is a budgeting rule: if the quadratic law is accurate, cutting the labeled data roughly in half should be compensated by shrinking depth or width by a factor near $\\sqrt{2}$, a prediction that could be checked on standard benchmarks without any new theory.","The covering-number route suggests the same machinery can be pushed to other alignment objectives, such as Wasserstein or correlation-alignment losses and norm-regularized feature extractors, where the depth exponent might differ; whether the quadratic-in-depth law is universal or an artifact of unconstrained parameterizations is a direct follow-up.","The paper itself notes in Remark 6 that norm constraints on weight matrices are expected to tighten the bounds, so a concrete test is whether spectrally regularized domain-adaptive networks beat the unconstrained quadratic sample law in practice.","Since the relatedness premise ties the certificate to loss-relevance of alignment, empirically estimating the constant $R$ on a benchmark would tell practitioners whether the quadratic law is the right budgeting tool there, especially under label shift with matched marginals."],"forward_implications":["Both MMD-based and adversarial domain-adaptive networks of width $d$ and depth $L$ require only on the order of $(d^2 L \\log(L/\\epsilon) + d^2 L^2 \\log d)/\\epsilon^2$ labeled source samples and total source and target samples to keep the target generalization gap at $O(\\epsilon)$, meaning sample needs grow quadratically in depth and width up to log factors.","In the adversarial setting the unlabeled-sample requirement is governed by the combined depth $L+K$ of the feature extractor plus domain discriminator, so a larger discriminator taxes the data budget just as a deeper classifier does.","Scaling the target-loss weight as $\\alpha = O(\\sqrt{M_t})$ avoids overfitting the model to a handful of target labels, so scarcity of labeled target data is handled by downweighting the target term rather than by acquiring more labels.","The bounds hold in terms of the empirical weighted loss and the empirical discrepancy computed on the aligned data, so they give a directly computable certificate of target performance rather than an abstract asymptotic statement.","Because the MMD and adversarial settings share the same scaling law and the same proof skeleton, the quadratic depth-width dependence appears to be a property of the alignment architecture itself rather than of the particular discrepancy objective."],"supporting_citations":[{"why":"Defines the MMD-based deep adaptation network architecture whose sample complexity Theorem 3 analyzes.","marker":"[14]"},{"why":"Defines the domain-adversarial architecture of feature extractor plus discriminator whose sample complexity Theorem 4 analyzes.","marker":"[21]"},{"why":"The classical source-target divergence analysis that this paper extends to settings where domains are aligned through learned transformations.","marker":"[33]"},{"why":"The single-domain neural-network sample complexity baseline used in Section 4.3 for comparison with the new quadratic depth-width law.","marker":"[44]"},{"why":"Supplies the definition of covering numbers on which all the complexity bounds rest.","marker":"[50]"},{"why":"Supplies the RKHS mean-embedding and kernel machinery used to model the MMD alignment setting.","marker":"[51]"},{"why":"The weighted source-plus-target-loss analysis whose alpha-scaling conclusions this paper refines into the explicit $O(\\sqrt{M_t})$ rate.","marker":"[55]"},{"why":"Yurinskii's exponential inequality for sums of random vectors is the engine behind Lemma 3's deviation bounds and hence the MMD sample-complexity result.","marker":"[76]"}],"fun_headline_variants":["Sample complexity bounds scale quadratically with network width and depth","Unified quadratic law for MMD and adversarial domain adaptation","For semi-supervised DA, target labels scale as square root of M_t","Depth-width squared sample complexity bound for domain alignment"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 1 (Eq. 3): a finite constant $R$ must exist such that, for every allowed transformation pair and hypothesis, the gap between the expected source loss and the expected target loss is at most $R$ times the distribution discrepancy between the aligned domains, and in the adversarial setting this is re-posed as Assumption 12 (Eq. 37), where the paper itself notes that the learned discriminator must be sufficiently well trained to certify poor alignment whenever the loss gap is large; if that relatedness condition fails, the target loss is no longer controlled by the source loss plus the discrepancy term, and Theorems 1 through 4 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sample complexity bounds scale quadratically with network width and depth","Unified quadratic law for MMD and adversarial domain adaptation","For semi-supervised DA, target labels scale as square root of M_t","Depth-width squared sample complexity bound for domain alignment"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2925,"prompt_tokens":1149,"completion_tokens":1776,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":1706}},"tokens_in":765,"tokens_out":1776,"duration_ms":17094,"temperature":1.0,"reasoning_tokens":1706,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:27:24.468214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a digit dataset and a target domain with the same input distribution but permuted class labels, so the marginal features can be aligned to near-zero MMD or discriminator confusion while the classification-loss gap stays large, and measure the ratio $|L^s - L^t|/D(f^s, f^t)$ across hypotheses; if that ratio can be driven arbitrarily large while the alignment loss is small, Assumption 1 or 12 fails and the constant $R$ in the bounds degenerates. Conversely, on benchmarks where the ratio stays bounded, the predicted quadratic growth of required sample sizes with depth and width, and the $\\alpha \\propto \\sqrt{M_t}$ schedule, are directly checkable by fitting the minimum $M_s$ and $N_s$ needed to hold a fixed target accuracy.","supporting_citations":[{"cited_title":"Learning transferable features with deep adaptation networks,","cited_arxiv_id":null,"evidence_quote":"Defines the MMD-based deep adaptation network architecture whose sample complexity Theorem 3 analyzes."},{"cited_title":"Domain-adversarial training of neural networks,","cited_arxiv_id":null,"evidence_quote":"Defines the domain-adversarial architecture of feature extractor plus discriminator whose sample complexity Theorem 4 analyzes."},{"cited_title":"Analysis of representations for domain adaptation,","cited_arxiv_id":null,"evidence_quote":"The classical source-target divergence analysis that this paper extends to settings where domains are aligned through learned transformations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The single-domain neural-network sample complexity baseline used in Section 4.3 for comparison with the new quadratic depth-width law."},{"cited_title":"On the Mathematical Foundations of Learning,","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of covering numbers on which all the complexity bounds rest."},{"cited_title":"A kernel two-sample test,","cited_arxiv_id":null,"evidence_quote":"Supplies the RKHS mean-embedding and kernel machinery used to model the MMD alignment setting."},{"cited_title":"A theory of learning from different domains,","cited_arxiv_id":null,"evidence_quote":"The weighted source-plus-target-loss analysis whose alpha-scaling conclusions this paper refines into the explicit $O(\\sqrt{M_t})$ rate."},{"cited_title":"Exponential inequalities for sums of random vectors,","cited_arxiv_id":null,"evidence_quote":"Yurinskii's exponential inequality for sums of random vectors is the engine behind Lemma 3's deviation bounds and hence the MMD sample-complexity result."}],"review_version":1}