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Model: google/gemini-3.1-pro-preview
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Task: optimization-pac-bayes-bound
Task: PAC-Bayes Generalization Bound Optimization
Research Question
Design a tighter PAC-Bayes generalization bound by optimizing the bound formulation, prior/posterior parameterization, and KL divergence estimation for stochastic neural networks.
Background
PAC-Bayes theory provides non-vacuous generalization bounds for stochastic classifiers. Given a prior distribution P over hypotheses (chosen before seeing data) and a posterior Q (learned from data), PAC-Bayes bounds certify that with high probability (1-delta), the true risk of a stochastic classifier sampled from Q is bounded.
The key components of a PAC-Bayes bound are:
- Empirical risk: estimated loss of the stochastic predictor on training data
- KL divergence: KL(Q||P) measuring complexity of the posterior relative to the prior
- Bound formula: how these terms combine to yield the final certificate
Standard bounds include:
- McAllester/Maurer: risk + sqrt(KL_term / (2n)) -- simple but loose
- Catoni/Lambda: risk/(1-lam/2) + KL_term/(nlam(1-lam/2)) -- tighter with tuned lambda
- Quadratic: (sqrt(risk + KL_term) + sqrt(KL_term))^2 -- better at low risk
The bound can be further tightened through:
- Optimizing the bound functional form (beyond classical inequalities)
- Better training objectives that minimize the bound directly
- Improved risk certificate evaluation (e.g., PAC-Bayes-kl inversion)
- Data-dependent prior construction
- Tighter KL estimation or alternative divergence measures
What to Implement
Implement the BoundOptimizer class in custom_pac_bayes.py. You must implement:
compute_bound(empirical_risk, kl, n, delta): The PAC-Bayes bound formulatrain_step(model, data, target, device, n_bound, delta): Training objectivecompute_risk_certificate(model, bound_loader, device, delta, mc_samples): Final certificate evaluation
Interface
model(x, sample=True/False): stochastic forward pass (sample=True) or posterior mean (sample=False)get_total_kl(model): sum of KL divergence across all probabilistic layersinv_kl(q, c): binary KL inversion -- find p such that KL(Ber(q)||Ber(p)) = ccompute_01_risk(model, loader, device, mc_samples): MC estimate of 0-1 risk- Available losses: F.nll_loss, F.cross_entropy on log_softmax outputs
Evaluation
The bound optimizer is tested on three settings:
- MNIST-FCN: 4-layer fully connected network (784-600-600-600-10) on MNIST
- MNIST-CNN: 4-layer CNN (2 conv + 2 fc) on MNIST
- FashionMNIST-CNN: Same CNN architecture on FashionMNIST
Primary metric: risk_certificate (0-1 loss PAC-Bayes bound) -- lower is better (tighter bound).
Training uses data-dependent priors: 50% of training data trains a deterministic prior, 50% evaluates the bound.
Hints
- The PAC-Bayes-kl bound (Langford-Seeger) is often tighter than closed-form alternatives for certificate evaluation
- The training objective need not be the sa
AssistantStep 1edit
Tool Input
1{
2 "after_line": 1,
3 "start_line": 460,
4 "filename": "PBB/custom_pac_bayes.py",
5 "content": "class BoundOptimizer:\n \"\"\"PAC-Bayes bound computation and posterior optimization.\n \n Uses an Optimal-Catoni bound analytically evaluated over Lambda,\n and implements data-dependent prior variance search via a valid PAC-Bayes union bound!\n \"\"\"\n\n def __init__(self, learning_rate=0.001, momentum=0.95, prior_sigma=0.03, pmin=1e-5):\n self.learning_rate = learning_rate\n self.momentum = momentum\n self.prior_sigma = prior_sigma\n self.pmin = pmin\n self.log_p_sigma = None\n self.ps_opt = None\n\n def compute_bound(self, empirical_risk, kl, n, delta):\n # Optimal analytical Lambda for Catoni PAC-Bayes bound\n c = (kl + math.log(2.0 * math.sqrt(n) / delta)) / n\n q = empirical_risk\n \n c_p = torch.clamp(c, min=1e-8)\n q_p = torch.clamp(q, min=1e-8)\n \n # Exact optimum of: min_{\\lambda} (q + c/\\lambda) / (1 - \\lambda/2)\n R = torch.sqrt(c_p**2 + 2.0 * q_p * c_p)\n bound = ((2.0 * q_p + c_p + R) ** 2) / (4.0 * q_p + 2.0 * c_p)\n return bound\n\n def train_step(self, model, data, target, device, n_bound, delta):\n prob_modules = [m for m in model.modules() if hasattr(m, 'prior_sigma')]\n M = len(prob_modules)\n \n if self.log_p_sigma is None:\n init_val = math.log(self.prior_sigma)\n self.log_p_sigma = torch.tensor([init_val] * M, device=device, requires_grad=True)\n self.ps_opt = torch.optim.Adam([self.log_p_sigma], lr=0.01)\n\n if self.log_p_sigma.grad is not None:\n self.ps_opt.step()\n self.ps_opt.zero_grad()\n \n with torch.no_grad():\n self.log_p_sigma.clamp_(math.log(0.0001), math.log(2.0))\n \n ps_vals = torch.exp(self.log_p_sigma)\n\n # Continously learn the per-layer prior sigmas anticipating the future union bound validation\n for i, m in enumerate(prob_modules):\n m.prior_sigma = ps_vals[i]\n\n output = model(data, sample=True)\n log_probs = F.log_softmax(output, dim=1)\n log_probs = torch.clamp(log_probs, min=math.log(self.pmin))\n nll = F.nll_loss(log_probs, target)\n\n kl = get_total_kl(model)\n \n K_layer = 1000\n K_total = float(K_layer) ** M\n bound = self.compute_bound(nll, kl, n_bound, delta / K_total)\n return bound\n\n def compute_risk_certificate(self, model, bound_loader, device, delta=0.025, mc_samples=1000):\n model.eval()\n n_bound = len(bound_loader.dataset)\n\n emp_risk_01 = compute_01_risk(model, bound_loader, device, mc_samples=mc_samples)\n\n total_nll = 0.0\n total_samples = 0\n with torch.no_grad():\n for data, target in bound_loader:\n data, target = data.to(device), target.to(device)\n output = model(data, sample=True)\n log_probs = F.log_softmax(output, dim=1)\n log_probs = torch.clamp(log_probs, min=math.log(self.pmin))\n nll = F.nll_loss(log_probs, target, reduction=\"sum\")\n total_nll += nll.item()\n total_samples += target.size(0)\n emp_nll = total_nll / total_samples\n\n # Grid Search for optimal prior_sigma per layer with valid union bound\n prob_modules = [m for m in model.modules() if hasattr(m, 'weight_mu')]\n M = len(prob_modules)\n K_layer = 1000\n sigmas = torch.logspace(-4, 0, K_layer).to(device)\n K_total = float(K_layer) ** M\n union_delta = delta / K_total\n \n best_kl = 0.0\n for m in prob_modules:\n best_m_kl = float('inf')\n best_m_sigma = None\n \n q_sigma_w = torch.log1p(torch.exp(m.weight_rho)).detach()\n w_diff_sq = (m.weight_mu.detach() - m.weight_prior_mu.detach()) ** 2\n \n q_sigma_b = torch.log1p(torch.exp(m.bias_rho)).detach()\n b_diff_sq = (m.bias_mu.detach() - m.bias_prior_mu.detach()) ** 2\n \n w_log_q = torch.log(q_sigma_w)\n b_log_q = torch.log(q_sigma_b)\n \n for p_sig in sigmas:\n p_var = p_sig.item() ** 2\n log_p_var = math.log(p_var)\n \n kl_w = 0.5 * (\n (q_sigma_w ** 2 + w_diff_sq) / p_var\n - 1.0 + log_p_var - 2.0 * w_log_q\n ).sum()\n \n kl_b = 0.5 * (\n (q_sigma_b ** 2 + b_diff_sq) / p_var\n - 1.0 + log_p_var - 2.0 * b_log_q\n ).sum()\n \n m_kl = (kl_w + kl_b).item()\n if m_kl < best_m_kl:\n best_m_kl = m_kl\n best_m_sigma = p_sig.item()\n \n m.prior_sigma = best_m_sigma\n best_kl += best_m_kl\n\n with torch.no_grad():\n dummy_data = next(iter(bound_loader))[0][:1].to(device)\n model(dummy_data, sample=True)\n\n c = (best_kl + math.log(2.0 * math.sqrt(n_bound) / union_delta)) / n_bound\n risk_cert_01 = inv_kl(emp_risk_01, c)\n\n emp_nll_t = torch.tensor(emp_nll)\n kl_t = torch.tensor(best_kl)\n ce_bound = self.compute_bound(emp_nll_t, kl_t, n_bound, union_delta).item()\n\n metrics = {\n \"empirical_01_risk\": emp_risk_01,\n \"empirical_nll\": emp_nll,\n \"kl_divergence\": best_kl,\n \"ce_bound\": ce_bound,\n }\n\n return risk_cert_01, metrics",
6 "end_line": 604,
7 "op": "replace"
8}Tool ResultStep 1
OK: Replaced lines 460..604 with 144 line(s) in PBB/custom_pac_bayes.py. Editable range: 460–603. [Current file: PBB/custom_pac_bayes.py | editable: 460–603 | total: 793 lines] 457| # ================================================================ 458| 459| 460 class BoundOptimizer: 461 """PAC-Bayes bound computation and posterior optimization. 462 463 Uses an Optimal-Catoni bound analytically evaluated over Lambda, 464 and implements data-dependent prior variance search via a valid PAC-Bayes union bound! 465 """ 466 467 def __init__(self, learning_rate=0.001, momentum=0.95, prior_sigma=0.03, pmin=1e-5): 468 self.learning_rate = learning_rate 469 self.momentum = momentum 470 self.prior_sigma = prior_sigma 471 self.pmin = pmin 472 self.log_p_sigma = None 473 self.ps_opt = None 474 475 def compute_bound(self, empirical_risk, kl, n, de
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1{
2 "after_line": 459,
3 "end_line": 604,
4 "content": "class DiffInvKL(torch.autograd.Function):\n @staticmethod\n def forward(ctx, q, c):\n q_val = min(max(q.item(), 1e-7), 1.0 - 1e-7)\n c_val = max(c.item(), 0.0)\n p_val = inv_kl(q_val, c_val)\n p_t = torch.tensor(p_val, device=q.device, dtype=q.dtype)\n ctx.save_for_backward(q, c, p_t)\n return p_t\n\n @staticmethod\n def backward(ctx, grad_output):\n q, c, p = ctx.saved_tensors\n q_val = torch.clamp(q, min=1e-7, max=1.0 - 1e-7)\n p_val = torch.clamp(p, min=q_val + 1e-6, max=1.0 - 1e-7)\n\n dp_dc = (p_val * (1.0 - p_val)) / (p_val - q_val + 1e-9)\n dkl_dq = torch.log((q_val * (1.0 - p_val)) / (p_val * (1.0 - q_val)))\n dp_dq = -dkl_dq * dp_dc\n\n return grad_output * dp_dq, grad_output * dp_dc\n\nclass BoundOptimizer:\n def __init__(self, learning_rate=0.001, momentum=0.95, prior_sigma=0.1, pmin=1e-5):\n self.learning_rate = learning_rate\n self.momentum = momentum\n self.prior_sigma = prior_sigma\n self.pmin = pmin\n\n def compute_bound(self, empirical_risk, kl, n, delta):\n c = (kl + math.log(2.0 * math.sqrt(n) / delta)) / n\n q = torch.clamp(empirical_risk, min=1e-7, max=0.999)\n return DiffInvKL.apply(q, c)\n\n def train_step(self, model, data, target, device, n_bound, delta):\n M = 2\n emp_risk = 0.0\n for _ in range(M):\n output = model(data, sample=True)\n probs = F.softmax(output, dim=1)\n p_correct = probs.gather(1, target.unsqueeze(1)).squeeze(1)\n emp_risk = emp_risk + (1.0 - p_correct).mean()\n \n emp_risk = emp_risk / M\n kl = get_total_kl(model)\n return self.compute_bound(emp_risk, kl, n_bound, delta)\n\n def compute_risk_certificate(self, model, bound_loader, device, delta=0.025, mc_samples=1000):\n model.eval()\n n_bound = len(bound_loader.dataset)\n\n emp_risk_01 = compute_01_risk(model, bound_loader, device, mc_samples=mc_samples)\n\n total_nll = 0.0\n total_samples = 0\n kl_total = None\n with torch.no_grad():\n for data, target in bound_loader:\n data, target = data.to(device), target.to(device)\n output = model(data, sample=True)\n log_probs = F.log_softmax(output, dim=1)\n log_probs = torch.clamp(log_probs, min=math.log(self.pmin))\n nll = F.nll_loss(log_probs, target, reduction=\"sum\")\n total_nll += nll.item()\n total_samples += target.size(0)\n if kl_total is None:\n kl_total = get_total_kl(model)\n\n emp_nll = total_nll / total_samples\n\n with torch.no_grad():\n dummy_data = next(iter(bound_loader))[0][:1].to(device)\n model(dummy_data, sample=True)\n kl = get_total_kl(model).item()\n\n c = (kl + math.log(2.0 * math.sqrt(n_bound) / delta)) / n_bound\n risk_cert_01 = inv_kl(emp_risk_01, c)\n\n emp_nll_t = torch.tensor(emp_nll, device=device)\n kl_t = torch.tensor(kl, device=device)\n ce_bound = self.compute_bound(emp_nll_t, kl_t, n_bound, delta).item()\n\n metrics = {\n \"empirical_01_risk\": emp_risk_01,\n \"empirical_nll\": emp_nll,\n \"kl_divergence\": kl,\n \"ce_bound\": ce_bound,\n }\n\n return risk_cert_01, metrics\n",
5 "filename": "PBB/custom_pac_bayes.py",
6 "op": "replace",
7 "start_line": 460
8}Tool ResultStep 1
OK: Replaced lines 460..604 with 90 line(s) in PBB/custom_pac_bayes.py. Editable range: 460–549. [Current file: PBB/custom_pac_bayes.py | editable: 460–549 | total: 739 lines] 457| # ================================================================ 458| 459| 460 class DiffInvKL(torch.autograd.Function): 461 @staticmethod 462 def forward(ctx, q, c): 463 q_val = min(max(q.item(), 1e-7), 1.0 - 1e-7) 464 c_val = max(c.item(), 0.0) 465 p_val = inv_kl(q_val, c_val) 466 p_t = torch.tensor(p_val, device=q.device, dtype=q.dtype) 467 ctx.save_for_backward(q, c, p_t) 468 return p_t 469 470 @staticmethod 471 def backward(ctx, grad_output): 472 q, c, p = ctx.saved_tensors 473 q_val = torch.clamp(q, min=1e-7, max=1.0 - 1e-7) 474 p_val = torch.clamp(p, min=q_val + 1e-6, max=1.0 - 1e-7) 475 476 dp_dc = (p_val * (1.0 - p_val)) /
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[Test #1] (2 tests remaining; call submit(n=1) to submit this result) ### mnist-fcn (scripts/mnist_fcn.sh) Training progress (last 5 steps): TRAIN_METRICS prior_epoch=10 loss=0.010002 accuracy=0.9971 TRAIN_METRICS posterior_epoch=5 train_obj=0.047469 kl=8.87 TRAIN_METRICS posterior_epoch=10 train_obj=0.047776 kl=10.06 TRAIN_METRICS posterior_epoch=15 train_obj=0.046266 kl=10.33 TRAIN_METRICS posterior_epoch=20 train_obj=0.047166 kl=9.23 Results (mnist-fcn): risk_certificate: 0.032692 test_error: 0.024900 kl_divergence: 9.230000 ce_bound: 0.171025 empirical_01_risk: 0.026600 ### mnist-cnn (scripts/mnist_cnn.sh) Training progress (last 5 steps): TRAIN_METRICS prior_epoch=10 loss=0.004733 accuracy=0.9990 TRAIN_METRICS posterior_epoch=5 train_obj=0.021482 kl=5.20 TRAIN_METRICS posterior_epoch=10 train_obj=0.021898 kl=6.04 TRAIN_METRICS posterior_epoch=15 train_obj=0.021326 kl=5.79 TRAIN_METRICS posterior_epoch=20 train_obj=0.021561 kl=6.01 Results (mnist-cnn): risk_certificat
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1{
2 "end_line": 495,
3 "filename": "PBB/custom_pac_bayes.py",
4 "content": " M = 4",
5 "after_line": 494,
6 "start_line": 495,
7 "op": "replace"
8}Tool ResultStep 3
OK: Replaced lines 495..495 with 1 line(s) in PBB/custom_pac_bayes.py. Editable range: 460–549. [Current file: PBB/custom_pac_bayes.py | editable: 460–549 | total: 739 lines] 457| # ================================================================ 458| 459| 460 class DiffInvKL(torch.autograd.Function): 461 @staticmethod 462 def forward(ctx, q, c): 463 q_val = min(max(q.item(), 1e-7), 1.0 - 1e-7) 464 c_val = max(c.item(), 0.0) 465 p_val = inv_kl(q_val, c_val) 466 p_t = torch.tensor(p_val, device=q.device, dtype=q.dtype) 467 ctx.save_for_backward(q, c, p_t) 468 return p_t 469 470 @staticmethod 471 def backward(ctx, grad_output): 472 q, c, p = ctx.saved_tensors 473 q_val = torch.clamp(q, min=1e-7, max=1.0 - 1e-7) 474 p_val = torch.clamp(p, min=q_val + 1e-6, max=1.0 - 1e-7) 475 476 dp_dc = (p_val * (1.0 - p_val)) /
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[Test #2] (1 test remaining; call submit(n=2) to submit this result) ### mnist-fcn (scripts/mnist_fcn.sh) [output file not found: /srv/home/bohanlyu/MLS-Bench/logs/optimization-pac-bayes-bound/google_gemini-3.1-pro-preview_20260405_234104/20260406_010831/group_1/mnist-fcn_s42.out] ### mnist-cnn (scripts/mnist_cnn.sh) [output file not found: /srv/home/bohanlyu/MLS-Bench/logs/optimization-pac-bayes-bound/google_gemini-3.1-pro-preview_20260405_234104/20260406_010831/group_1/mnist-cnn_s42.out] ### fmnist-cnn (scripts/fmnist_cnn.sh) [output file not found: /srv/home/bohanlyu/MLS-Bench/logs/optimization-pac-bayes-bound/google_gemini-3.1-pro-preview_20260405_234104/20260406_010831/group_1/fmnist-cnn_s42.out]
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Undo complete: Restored: PBB/custom_pac_bayes.py
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2 "n": 1
3}Tool ResultStep 6
[submit] Submitting result from test #1 as final. ### mnist-fcn (scripts/mnist_fcn.sh) Training progress (last 5 steps): TRAIN_METRICS prior_epoch=10 loss=0.010002 accuracy=0.9971 TRAIN_METRICS posterior_epoch=5 train_obj=0.047469 kl=8.87 TRAIN_METRICS posterior_epoch=10 train_obj=0.047776 kl=10.06 TRAIN_METRICS posterior_epoch=15 train_obj=0.046266 kl=10.33 TRAIN_METRICS posterior_epoch=20 train_obj=0.047166 kl=9.23 Results (mnist-fcn): risk_certificate: 0.032692 test_error: 0.024900 kl_divergence: 9.230000 ce_bound: 0.171025 empirical_01_risk: 0.026600 ### mnist-cnn (scripts/mnist_cnn.sh) Training progress (last 5 steps): TRAIN_METRICS prior_epoch=10 loss=0.004733 accuracy=0.9990 TRAIN_METRICS posterior_epoch=5 train_obj=0.021482 kl=5.20 TRAIN_METRICS posterior_epoch=10 train_obj=0.021898 kl=6.04 TRAIN_METRICS posterior_epoch=15 train_obj=0.021326 kl=5.79 TRAIN_METRICS posterior_epoch=20 train_obj=0.021561 kl=6.01 Results (mnist-cnn): risk_certificate: 0.015583 test_e