================================================================================ CLAUDE CODE SESSION EXPORT File: ale_dp_selftest.txt Folder: c:\Projects\Agents Last Exam Date: 2026-08-11 Model: Opus 5 (1M context) Topic: Renyi divergence bound for noisy projected GD with clipped, Holder-continuous gradients (hidden-state / last-iterate analysis) ================================================================================ -------------------------------------------------------------------------------- USER -------------------------------------------------------------------------------- Help solve the following question: Let K ⊂ R^d be a closed convex set with diameter D. Consider W_{t+1} = Π_K [ W_t − (η/n) Σ_{i=1}^n Π_{B_K}[ ∇ℓ_i(W_t) ] + G_t ], G_t ~ N(0, σ² I), for t = 0,...,T−1, where Π_{B_K} denotes projection onto the ℓ2-ball of radius K. Let W'_t be the corresponding process when exactly one loss term ℓ_i is replaced by ℓ'_i, with W_0 = W'_0. Assume that for every loss function involved, ‖∇ℓ_i(x) − ∇ℓ_i(y)‖ ≤ L‖x − y‖^λ, 0 < λ ≤ 1, for all x, y ∈ K. Only the final iterate W_T is released. For any α > 1, derive the tightest upper bound you can on D_α( L(W_T) ‖ L(W'_T) ). Your final result should be explicit enough to evaluate numerically for arbitrary (T, n, η, K, L, λ, σ, D, α). -------------------------------------------------------------------------------- ASSISTANT -------------------------------------------------------------------------------- I'll work this out with the shifted-Renyi / privacy-amplification-by-iteration machinery, then verify the resulting formula numerically against a brute-force dynamic program. [Tool use: checked Python/NumPy availability -> numpy 2.0.2] [Tool use: wrote scratchpad script holder_renyi.py; ran it; ran a second cross-check focused on parameter regimes where the diameter reset is active] === Setup: three per-step constants === Write the update as a noisy, projected map. With F(x) = (1/n) Σ_i Π_{B_K}[∇ℓ_i(x)], φ(x) = x − ηF(x), we have W_{t+1} = Π_K[φ(W_t) + G_t], and W'_{t+1} = Π_K[φ'(W'_t) + G_t] with φ' built from the neighbouring dataset. Everything below is driven by three quantities. (1) SENSITIVITY. Only one summand differs and both are clipped to norm ≤ K: s := sup_{x∈K} ‖φ(x) − φ'(x)‖ ≤ (η/n)·2K = 2ηK/n. (2) EXPANSION. Projection onto B_K is 1-Lipschitz, so clipping *preserves* the Holder modulus: ‖Π_{B_K}∇ℓ_i(x) − Π_{B_K}∇ℓ_i(y)‖ ≤ L‖x−y‖^λ, hence ‖F(x) − F(y)‖ ≤ L‖x−y‖^λ and ‖φ(x) − φ(y)‖ ≤ c(‖x−y‖), c(z) := z + ηL z^λ. c is increasing, and c(z) ≥ z: with only Holder continuity the map is EXPANSIVE, never contractive. This is the crux — it kills the usual convex/smooth amplification-by-iteration gain, and the entire improvement over composition will come from D. (3) DIAMETER CEILING. Coupling the two chains with the SAME G_t and using that Π_K is 1-Lipschitz, ‖W_{t+1} − W'_{t+1}‖ ≤ c(‖W_t − W'_t‖) + s. So W_∞( L(W_t), L(W'_t) ) ≤ R_t where R_0 = 0, R_{t+1} = min{ D, c(R_t) + s }. (R_t ≤ D because both iterates live in K.) === Toolkit: shifted Renyi divergence === For α > 1 define D_α^{(z)}(μ‖ν) := inf_{ μ' : W_∞(μ,μ') ≤ z } D_α(μ'‖ν) (Feldman-Mironov-Talwar-Thakurta '18). Four facts: (L1) Shift reduction. D_α^{(z)}( μ*N(0,σ²I) ‖ ν*N(0,σ²I) ) ≤ D_α^{(z+a)}(μ‖ν) + α a²/(2σ²), any a ≥ 0. (L2) Data processing. For 1-Lipschitz Π: D_α^{(z)}(Π_#μ ‖ Π_#ν) ≤ D_α^{(z)}(μ‖ν). (L3) Expansion + drift. D_α^{(c(z)+s)}( φ_#μ ‖ φ'_#ν ) ≤ D_α^{(z)}(μ‖ν). Proof: take μ' attaining the inf; D_α(φ'_#μ' ‖ φ'_#ν) ≤ D_α(μ'‖ν) by post-processing, and for a W_∞-coupling (X,X'), ‖φ(X) − φ'(X')‖ ≤ ‖φ(X) − φ(X')‖ + s ≤ c(z) + s. □ (L4) Free entry. D_α^{(z)}(μ_t‖ν_t) = 0 whenever z ≥ R_t (take μ' = ν_t). === One-step recursion === Chaining (L2) -> (L1) -> (L3): for any a_{t+1} ≥ 0 and any shifts with z_{t+1} + a_{t+1} ≥ c(z_t) + s, D_α^{(z_{t+1})}(μ_{t+1}‖ν_{t+1}) ≤ D_α^{(z_t)}(μ_t‖ν_t) + α a_{t+1}²/(2σ²). === Master bound (tightest in this framework) === Every admissible chain must start at a FREE entry point (L4) and end at shift 0 at time T. Hence, with a_j := c(z_{j-1}) + s − z_j, D_α( L(W_T) ‖ L(W'_T) ) ≤ (α / 2σ²) · min_{1 ≤ τ ≤ T} C( τ, R_{T−τ} ) C(τ,R) = min Σ_{j=1}^{τ} ( c(z_{j-1}) + s − z_j )² z_0=R, z_τ=0 z_j ≥ 0, c(z_{j-1}) + s − z_j ≥ 0 Equivalently as a DP: V_0 ≡ 0, V_{t+1}(z) = min_{z'≥0} [ V_t(z') + (α/2σ²)·(( c(z') + s − z )_+)² ], capped by V_t(z) = 0 for z ≥ R_t; the answer is V_T(0). Note τ = T gives R_0 = 0, z ≡ 0, a_j ≡ s, i.e. THE ORDINARY COMPOSITION BOUND α T s² / (2σ²) = 2 α T η² K² / (n² σ²) is always one of the candidates. === Explicit closed form (this is what you evaluate) === Take the linear schedule z_j = R(1 − j/τ), which is always feasible and — checked numerically — within 0.2%–4% of the DP optimum. With S_p(τ) := Σ_{m=1}^τ m^p: D_α ≤ (α / 2σ²) · min_{1 ≤ τ ≤ T} [ τ (R/τ + s)² + 2 (R/τ + s) · ηL R^λ S_λ(τ) / τ^λ + (ηL R^λ)² S_{2λ}(τ) / τ^{2λ} ] with R = R_{T−τ} with s = 2ηK/n, c(z) = z + ηLz^λ, R_t = min{D, c(R_{t-1}) + s}, R_0 = 0. Fully explicit in (T, n, η, K, L, λ, σ, D, α); dimension-free; O(T) to evaluate. --- Large-T saturation (closed form, no minimisation) --- Put R = D, use S_p(τ) ≈ τ^{p+1}/(p+1) and optimise τ continuously — the optimum is τ* = D / sqrt( (s+β)² + κ² ): D_α ≤ (αD/σ²) · [ (s+β) + sqrt( (s+β)² + κ² ) ], β = ηL D^λ / (1+λ), κ = λ ηL D^λ / ( (1+λ) sqrt(1+2λ) ), which relaxes to the memorable, T-INDEPENDENT statement D_α ≤ (2αD/σ²) · ( 2ηK/n + ηL D^λ ) for every T. CROSSOVER. Composition wins until roughly T* ≈ 4D(s+β)/s²; with L small this is T* ≈ 4D/s = 2Dn/(ηK). So the summary is D_α ≲ (α/2σ²) · min{ T s² (composition) , 2D[ (s+β) + sqrt((s+β)² + κ²) ] } (saturation) --- Refinement for λ = 1 --- Then c(z) = ρz with ρ = 1 + ηL, and C(τ,R) is a quadratic program solved exactly by Cauchy-Schwarz (a_j ∝ ρ^{τ−j}): C(τ,R) = [ (ρ²−1) / (ρ^{2τ}−1) ] · [ ρ^τ R + s(ρ^τ−1)/(ρ−1) ]² (use it when the induced z_j stay ≥ 0; else the linear schedule). === Two remarks on tightness === 1. WITHOUT D THE BOUND CANNOT BEAT COMPOSITION. If z_0 = z_τ = 0, then since c(z) ≥ z and x -> x² is convex, Σ_j ( c(z_{j-1}) + s − z_j )² ≥ Σ_j ( z_{j-1} − z_j + s )² ≥ τ s². So for merely Holder (non-convex) losses, hidden-state analysis yields NOTHING on an unbounded domain; all of the gain is compactness of K. This is the honest answer to "tightest you can". 2. The saturation shape αDs/σ² matches the known lower-bound order in the smooth convex case (Altschuler-Talwar), so the T-independent regime is not improvable in general; only constants are. TO (ε,δ)-DP: ε = min_{α>1} [ ε_α + log((α−1)/α) − (log δ + log α)/(α−1) ], and ε_α above is linear in α, so this minimisation is one-dimensional and trivial. === Reference implementation + verification === Sample output, η=0.5, K=1, n=10, L=0.1, λ=1/2, σ=1, D=1, α=2 (so s = 0.1): T composition closed form (τ*) DP (tightest) saturation 20 0.200 0.200 (20) 0.200 0.534 50 0.500 0.500 (50) 0.500 0.534 200 2.00 0.546 (8) 0.545 0.534 600 6.00 0.546 (8) 0.545 0.534 The closed form reproduces the DP optimum to within 0.2% here (worst case seen across the parameter grids run: 4%, at λ=1), tracks composition exactly in the small-T regime, and flattens at the predicted saturation level. Additional runs (from the script's __main__ block): smooth, λ=1: η=0.5, K=1, n=100, L=0.1, σ=1, D=1, α=2 T composition closed form τ* DP (grid) saturation 10 0.001 0.001 10 0.001 0.14572 100 0.01 0.01 100 0.01 0.14572 1000 0.1 0.1 1000 0.1 0.14572 10000 1 0.14935 27 n/a 0.14572 100000 10 0.14935 27 n/a 0.14572 (T=3000 DP cross-check: closed form 0.149354 vs DP 0.143505, ratio 1.041) Holder, λ=1/2: η=0.1, K=1, n=100, L=0.1, σ=1, D=1, α=2 T composition closed form τ* DP (grid) saturation 10 4e-05 4e-05 10 4e-05 0.035296 100 0.0004 0.0004 100 0.0004 0.035296 1000 0.004 0.004 1000 0.004 0.035296 10000 0.04 0.035452 112 n/a 0.035296 100000 0.4 0.035452 112 n/a 0.035296 L=0 (pure clipping): η=0.5, K=1, n=100, λ=1/2, σ=1, D=1, α=2 T composition closed form τ* DP (grid) saturation 10 0.001 0.001 10 0.001 0.04 100 0.01 0.01 100 0.01 0.04 1000 0.1 0.04 100 0.040003 0.04 10000 1 0.04 100 n/a 0.04 100000 10 0.04 100 n/a 0.04 rough, λ=1/4: η=0.05, K=1, n=500, L=1, σ=1, D=2, α=4 T composition closed form τ* DP (grid) saturation 10 8e-07 8e-07 10 8e-07 0.77211 100 8e-06 8e-06 100 8e-06 0.77211 1000 8e-05 8e-05 1000 8e-05 0.77211 10000 0.0008 0.0008 10000 n/a 0.77211 100000 0.008 0.008 100000 n/a 0.77211 η=0.2, K=1, n=20, L=0.5, λ=1/4, σ=0.7, D=3, α=8 T=50 comp=0.16327 closed=0.163265 (τ*=50) DP=0.163265 ratio=1.0000 T=200 comp=0.65306 closed=0.653061 (τ*=200) DP=0.653061 ratio=1.0000 T=800 comp=2.6122 closed=2.61224 (τ*=800) DP=2.61224 ratio=1.0000 saturation = 12.363 ================================================================================ APPENDIX: holder_renyi.py (reference implementation, verbatim) ================================================================================ """ Renyi bound for noisy projected full-batch GD with clipped, Holder-continuous gradients. Per-step objects: s = 2*eta*K/n (sup_x |phi(x) - phi'(x)|, replace-one) c(z) = z + eta*L*z**lam (expansion of the GD map on shifts) R_t = min(D, r_t), r_0 = 0, r_{t+1} = c(r_t) + s (free W_inf reset level) Bound: D_alpha <= (alpha/(2 sigma^2)) * min_tau C(tau, R_{T-tau}) where C(tau,R) = min over shift schedules z_0=R,...,z_tau=0 of sum_t (c(z_{t-1})+s-z_t)^2. """ import numpy as np def per_step(eta, K, n, L, lam): s = 2.0 * eta * K / n c = lambda z: z + eta * L * np.power(z, lam) return s, c def reset_levels(T, eta, K, n, L, lam, D): """R_t = min(D, r_t) with r_{t+1} = c(r_t) + s, r_0 = 0.""" s, c = per_step(eta, K, n, L, lam) R = np.zeros(T + 1) r = 0.0 for t in range(1, T + 1): r = min(D, c(r) + s) R[t] = r return R def C_linear(tau, R, eta, K, n, L, lam, Slam, S2lam): """Cost of the linear schedule z_j = R(1 - j/tau), closed form.""" s = 2.0 * eta * K / n A = R / tau + s B = eta * L * R ** lam return (tau * A * A + 2.0 * A * B * tau ** (-lam) * Slam[tau] + B * B * tau ** (-2.0 * lam) * S2lam[tau]) def bound_closed_form(T, eta, K, n, L, lam, sigma, D, alpha, return_tau=False): R = reset_levels(T, eta, K, n, L, lam, D) m = np.arange(0, T + 1, dtype=float) Slam = np.concatenate(([0.0], np.cumsum(m[1:] ** lam))) S2lam = np.concatenate(([0.0], np.cumsum(m[1:] ** (2 * lam)))) best, best_tau = np.inf, None for tau in range(1, T + 1): val = C_linear(tau, R[T - tau], eta, K, n, L, lam, Slam, S2lam) if val < best: best, best_tau = val, tau out = alpha / (2 * sigma ** 2) * best return (out, best_tau) if return_tau else out def bound_saturated(eta, K, n, L, lam, sigma, D, alpha): """tau-continuous optimum of the linear schedule with R = D (large-T limit).""" s = 2.0 * eta * K / n B = eta * L * D ** lam beta = B / (lam + 1.0) kappa = lam * B / ((lam + 1.0) * np.sqrt(2 * lam + 1.0)) return alpha * D / sigma ** 2 * ((s + beta) + np.sqrt((s + beta) ** 2 + kappa ** 2)) def bound_composition(T, eta, K, n, sigma, alpha): return 2.0 * alpha * T * eta ** 2 * K ** 2 / (n ** 2 * sigma ** 2) def bound_dp(T, eta, K, n, L, lam, sigma, D, alpha, grid=600): """Brute-force DP over a grid of shift values -- the tightest value the framework can give (up to discretisation). V_t(z)=0 for z >= R_t.""" s, c = per_step(eta, K, n, L, lam) R = reset_levels(T, eta, K, n, L, lam, D) z = np.linspace(0.0, D, grid) cz = c(z) # cost[i, j] = ((c(z_i) + s - z_j)_+)^2 (from shift z_i at time t to z_j at t+1) cost = np.clip(cz[:, None] + s - z[None, :], 0.0, None) ** 2 * (alpha / (2 * sigma ** 2)) V = np.where(z >= R[0] - 1e-15, 0.0, np.inf) # t = 0: R_0 = 0 -> V_0 == 0 everywhere for t in range(1, T + 1): V = (V[:, None] + cost).min(axis=0) V = np.minimum(V, np.where(z >= R[t] - 1e-12, 0.0, np.inf)) return V[0] if __name__ == "__main__": cases = [ dict(name="smooth, lam=1", eta=0.5, K=1.0, n=100, L=0.1, lam=1.0, sigma=1.0, D=1.0, alpha=2.0), dict(name="Holder, lam=1/2", eta=0.1, K=1.0, n=100, L=0.1, lam=0.5, sigma=1.0, D=1.0, alpha=2.0), dict(name="L=0 (pure clipping)", eta=0.5, K=1.0, n=100, L=0.0, lam=0.5, sigma=1.0, D=1.0, alpha=2.0), dict(name="rough, lam=1/4", eta=0.05, K=1.0, n=500, L=1.0, lam=0.25, sigma=1.0, D=2.0, alpha=4.0), ] for cs in cases: nm = cs.pop("name") print(f"\n=== {nm}: {cs}") print(f"{'T':>7} {'composition':>13} {'closed form':>13} {'tau*':>6} {'DP (grid)':>13} {'saturation':>13}") sat = bound_saturated(**cs) for T in [10, 100, 1000, 10000, 100000]: comp = bound_composition(T, cs['eta'], cs['K'], cs['n'], cs['sigma'], cs['alpha']) cf, tau = bound_closed_form(T, **cs, return_tau=True) dp = bound_dp(T, **cs) if T <= 1000 else float('nan') print(f"{T:>7} {comp:13.5g} {cf:13.5g} {tau:>6} {dp:13.5g} {sat:13.5g}") ================================================================================ END OF EXPORT ================================================================================