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diff --git a/notebooks/recursive_reasoning_chaos.ipynb b/notebooks/recursive_reasoning_chaos.ipynb index 097f9b6..1f00628 100644 --- a/notebooks/recursive_reasoning_chaos.ipynb +++ b/notebooks/recursive_reasoning_chaos.ipynb @@ -2,33 +2,32 @@ "cells": [ { "cell_type": "markdown", - "id": "6c32c5e8", + "id": "f991b29d", "metadata": {}, "source": [ - "# Recursive Reasoning Failures are Chaotic — and it's *transient chaos*\n", + "# Recursive Reasoning Failures are Chaotic\n", "\n", - "Small recursive reasoners (HRM, TRM) iterate a latent state to solve puzzles (Sudoku, Maze).\n", - "Measured along the inference trajectory, **failed examples are more chaotic** (higher finite-time\n", - "Lyapunov exponent / latent drift) than successful ones, in the *same* trained network.\n", + "Small recursive reasoners (HRM, TRM) iterate a latent state to solve a puzzle (Sudoku) before\n", + "emitting an answer. **The core finding:** measured along the recurrent inference trajectory, the\n", + "**leading finite-time Lyapunov exponent (λ₁) is higher on failed examples than on successful ones**\n", + "— in the *same* trained network. Failure is locally more chaotic.\n", "\n", - "This notebook lets you reproduce and play with the mechanism:\n", - "1. **Toy model** (pure numpy, no GPU) — *transient chaos*: chaotic search of latent space until the\n", - " trajectory escapes into the solution basin. Failures = not-yet-escaped trajectories.\n", - "2. **Real trained model** loaded from HuggingFace (`blackhao0426/recursive-reasoning-chaos`).\n", - "3. **Extended rollout** — run the recurrence far beyond its training budget. Both architectures'\n", - " failures sit on a chaotic *saddle* (transient chaos), not a wrong fixed point — they just escape\n", - " at very different rates: **TRM** failures mostly escape and self-correct given enough compute;\n", - " **HRM** failures are far more strongly trapped (most keep churning).\n", - "4. **Basin accessibility** — restart a trapped puzzle from perturbed initial latent states. A small\n", - " kick frees most of TRM's (IC-determined, large basin); a hard core of HRM's never escapes any\n", - " nearby initial condition (input-determined).\n", + "This notebook, top to bottom:\n", + "1. **Load** a trained model from HuggingFace (`blackhao0426/recursive-reasoning-chaos`).\n", + "2. **The result** — compute λ₁ per example (Benettin / JVP along the trajectory) and show the\n", + " success-vs-failure separation (histogram + AUC). *This is the headline.*\n", + "3. **Why** — run the recurrence far past its training budget: failures are a *transient* — they\n", + " escape the chaotic set and self-correct given enough compute (TRM), or stay trapped much longer\n", + " (HRM). Neither settles to a wrong fixed point.\n", + "4. **Basin accessibility** — restart from a perturbed initial state: is a failure input-determined\n", + " or initial-condition-determined?\n", "\n", "Companion analysis repo: `github.com/YurenHao0426/recursive-reasoning-dynamics`." ] }, { "cell_type": "markdown", - "id": "9161f5cf", + "id": "bb82e0a8", "metadata": {}, "source": [ "## 0. Setup" @@ -37,82 +36,32 @@ { "cell_type": "code", "execution_count": null, - "id": "2034b179", + "id": "89a96776", "metadata": {}, "outputs": [], "source": [ - "# minimal deps; torch+einops+pydantic are enough to load these models (TRM-Sudoku is MLP-mixer,\n", - "# no FlashAttention needed -> runs on any GPU, even CPU).\n", - "%pip install -q torch einops pydantic huggingface_hub numpy matplotlib\n", + "%pip install -q torch einops pydantic huggingface_hub numpy matplotlib tqdm\n", "import numpy as np, matplotlib.pyplot as plt, torch\n", + "from tqdm.auto import tqdm\n", "print(\"torch\", torch.__version__, \"| cuda\", torch.cuda.is_available())" ] }, { "cell_type": "markdown", - "id": "fe43cb07", + "id": "bbd25841", "metadata": {}, "source": [ - "## 1. The toy model — transient chaos (no GPU, runs in seconds)\n", + "## 1. Load a trained model from HuggingFace\n", "\n", - "A trajectory chaotically *searches* `[0,1]` (logistic map, λ=ln2≈+0.69) until it lands within `eps`\n", - "of the solution `s` (the \"puzzle\"), then it converges (λ=ln0.5<0). At a fixed readout time `T`:\n", - "**captured = success** (FTLE low), **still searching = failure** (FTLE high). The escape time is\n", - "~geometric (chaotic-saddle signature) and the FTLE separation is purely a *finite-time* effect —\n", - "it vanishes as `T→∞` because everyone eventually escapes." + "Downloads model code + checkpoint + a 2000-puzzle test set from `blackhao0426/recursive-reasoning-chaos`.\n", + "`MODEL` ∈ {`trm_sudoku`, `hrm_sudoku`}. **TRM is MLP-only → runs on a laptop CPU.** To switch\n", + "models, change `MODEL` and **restart the kernel** (the two ship same-named `models` packages)." ] }, { "cell_type": "code", "execution_count": null, - "id": "6593c881", - "metadata": {}, - "outputs": [], - "source": [ - "def run_toy(n=20000, T=16, eps=0.04, seed=0):\n", - " rg = np.random.default_rng(seed)\n", - " s = rg.uniform(0.15, 0.85, n); x = rg.uniform(0, 1, n)\n", - " captured = np.zeros(n, bool); logd = np.zeros(n)\n", - " for t in range(T):\n", - " search = ~captured\n", - " ld = np.where(search, np.log(np.abs(4*(1-2*x))+1e-12), np.log(0.5))\n", - " xn = np.where(search, 4*x*(1-x), s + 0.5*(x-s))\n", - " captured |= search & (np.abs(xn-s) < eps); x = xn; logd += ld\n", - " ftle = logd / T\n", - " success = captured & (np.abs(x-s) < 0.05)\n", - " return ftle, success\n", - "\n", - "def auc(score, y):\n", - " p, n = score[y==1], score[y==0]; a=np.concatenate([p,n]); o=np.argsort(a)\n", - " r=np.empty(len(a)); r[o]=np.arange(1,len(a)+1)\n", - " return (r[:len(p)].sum()-len(p)*(len(p)+1)/2)/(len(p)*len(n))\n", - "\n", - "ftle, succ = run_toy(T=16)\n", - "print(f\"success rate {succ.mean():.2f} | FTLE success {np.median(ftle[succ]):+.3f} vs failure {np.median(ftle[~succ]):+.3f}\")\n", - "print(f\"AUC(-FTLE -> success) = {auc(-ftle, succ.astype(int)):.3f} (failure more chaotic)\")\n", - "fig,ax=plt.subplots(1,2,figsize=(11,4))\n", - "b=np.linspace(-0.5,0.75,50)\n", - "ax[0].hist(ftle[succ],b,alpha=.6,color='g',density=True,label='success'); ax[0].hist(ftle[~succ],b,alpha=.6,color='r',density=True,label='failure')\n", - "ax[0].set_title('toy: failure more chaotic'); ax[0].set_xlabel('finite-time Lyapunov exp'); ax[0].legend()\n", - "Ts=[4,8,16,32,64,128,256]; A=[auc(-run_toy(T=T)[0],run_toy(T=T)[1].astype(int)) for T in Ts]; R=[run_toy(T=T)[1].mean() for T in Ts]\n", - "ax[1].plot(Ts,A,'o-',label='AUC(-FTLE->success)'); ax[1].plot(Ts,R,'s--',label='success rate'); ax[1].set_xscale('log')\n", - "ax[1].set_xlabel('readout time T'); ax[1].set_title('finite-time: separation vanishes as T->inf'); ax[1].legend(); plt.tight_layout(); plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "aee64679", - "metadata": {}, - "source": [ - "## 2. Load a trained model from HuggingFace\n", - "\n", - "Downloads the model code + checkpoint + config from `blackhao0426/recursive-reasoning-chaos`. `MODEL` ∈ {`trm_sudoku`, `hrm_sudoku`}." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "5f5f69ff", + "id": "f6a83ba0", "metadata": {}, "outputs": [], "source": [ @@ -123,8 +72,6 @@ "HF_REPO = \"blackhao0426/recursive-reasoning-chaos\"\n", "MODEL = \"trm_sudoku\" # or \"hrm_sudoku\"\n", "root = Path(snapshot_download(HF_REPO))\n", - "# TRM and HRM ship separate `models/` packages -> put the right one on the path.\n", - "# (To switch MODEL, restart the kernel: Python caches the `models` package name.)\n", "sys.path.insert(0, str(root / (\"code_trm\" if MODEL.startswith(\"trm\") else \"code_hrm\")))\n", "\n", "cfg = yaml.safe_load((root / MODEL / \"all_config.yaml\").read_text())\n", @@ -147,26 +94,113 @@ }, { "cell_type": "markdown", - "id": "cea384ce", + "id": "086e411f", "metadata": {}, "source": [ - "## 3. Extended rollout — the mechanism\n", + "## 2. The core result — failures are more chaotic (leading FTLE / λ₁)\n", + "\n", + "For each puzzle we run the recurrence for the 16-segment inference budget and propagate one tangent\n", + "vector through every module update (forward-mode JVP), renormalizing each step and accumulating the\n", + "log-growth (Benettin's method for the largest exponent). λ₁ = mean log-growth per module-evaluation.\n", + "Then split by outcome at segment 16. **Failures sit at higher λ₁** — they are locally expanding /\n", + "chaotic; successes have collapsed toward the solution. `AUC(−λ₁ → success)` near 1 = clean separation.\n", "\n", - "Run the recurrence `N_SEG` segments (far past the 16-segment training budget) and watch the fate of\n", - "trajectories that fail at segment 16. Re-run cell 2 with `MODEL=\"hrm_sudoku\"` to see the contrast." + "The JVP uses the fact that the update is `module(a, b) = layers(a + b)`, so a perturbation of the\n", + "combined input can be fed through one slot. (GPU: ~1 min. Laptop/CPU with TRM: a few min — lower `n`.)" ] }, { "cell_type": "code", "execution_count": null, - "id": "5d7ec0ce", + "id": "791d02dc", + "metadata": {}, + "outputs": [], + "source": [ + "import torch.autograd.functional as AF\n", + "from contextlib import nullcontext\n", + "try: # HRM attention JVP needs the math SDP backend (no FlashAttn double-backward)\n", + " from torch.nn.attention import sdpa_kernel, SDPBackend\n", + " MATHCTX = lambda: sdpa_kernel(SDPBackend.MATH)\n", + "except Exception:\n", + " MATHCTX = nullcontext\n", + "\n", + "def auc(score, y):\n", + " p, n = score[y==1], score[y==0]\n", + " if len(p)==0 or len(n)==0: return float('nan')\n", + " a=np.concatenate([p,n]); o=np.argsort(a); r=np.empty(len(a)); r[o]=np.arange(1,len(a)+1)\n", + " return (r[:len(p)].sum()-len(p)*(len(p)+1)/2)/(len(p)*len(n))\n", + "\n", + "def leading_ftle(inp, lab, pid, n=128, n_seg=16, seed=0):\n", + " rng=np.random.default_rng(seed); idx=rng.choice(len(inp), n, replace=False)\n", + " pe=inner.puzzle_emb_len; sf=inner.config.seq_len+pe; hid=inner.config.hidden_size; D=sf*hid; B=n\n", + " is_hrm=hasattr(inner,\"H_level\") and getattr(inner,\"H_level\",None) is not None\n", + " Hmod=inner.H_level if is_hrm else inner.L_level # weight-tied TRM reuses L_level\n", + " X=torch.tensor(inp[idx].astype(np.int32),device=dev); Y=torch.tensor(lab[idx].astype(np.int32),device=dev)\n", + " P=torch.tensor(pid[idx].astype(np.int32),device=dev)\n", + " si=dict(cos_sin=inner.rotary_emb() if hasattr(inner,\"rotary_emb\") else None)\n", + " g=torch.Generator(device=dev).manual_seed(seed)\n", + " jvp=lambda f,x,v: AF.jvp(f, x, v=v, create_graph=False, strict=False)\n", + " def renorm(vH,vL):\n", + " nrm=torch.sqrt(vH.pow(2).sum(1,keepdim=True)+vL.pow(2).sum(1,keepdim=True)).clamp_min(1e-30)\n", + " return vH/nrm, vL/nrm, nrm.squeeze(1)\n", + " with MATHCTX():\n", + " emb=inner._input_embeddings(X,P); m=Y>0\n", + " zH=inner.H_init.unsqueeze(0).expand(B,sf,hid).clone().to(inner.forward_dtype)\n", + " zL=inner.L_init.unsqueeze(0).expand(B,sf,hid).clone().to(inner.forward_dtype)\n", + " vH=torch.randn(B,D,device=dev,generator=g); vL=torch.randn(B,D,device=dev,generator=g)\n", + " vH,vL,_=renorm(vH,vL); logsum=torch.zeros(B,device=dev); nstep=0\n", + " for seg in tqdm(range(n_seg), desc=\"FTLE (segments)\"):\n", + " with torch.enable_grad():\n", + " zH,zL=zH.detach(),zL.detach()\n", + " for _h in range(inner.config.H_cycles):\n", + " for _l in range(inner.config.L_cycles):\n", + " vc=(vH+vL).reshape(B,sf,hid).to(inner.forward_dtype)\n", + " zL,Dv=jvp(lambda z: inner.L_level(z, zH+emb, **si), zL, vc)\n", + " vL=Dv.reshape(B,D).float(); vH,vL,grow=renorm(vH,vL); logsum+=grow.log(); nstep+=1\n", + " vc=(vH+vL).reshape(B,sf,hid).to(inner.forward_dtype)\n", + " zH,Dv=jvp(lambda z: Hmod(z, zL, **si), zH, vc)\n", + " vH=Dv.reshape(B,D).float(); vH,vL,grow=renorm(vH,vL); logsum+=grow.log(); nstep+=1\n", + " ftle=(logsum/nstep).cpu().numpy()\n", + " ok=(((inner.lm_head(zH)[:,pe:].float().argmax(-1)==Y)|~m).all(-1)).cpu().numpy()\n", + " return ftle, ok\n", + "\n", + "ftle, succ = leading_ftle(inp, lab, pid, n=128)\n", + "print(f\"success rate {succ.mean():.2f} | median λ1 success {np.median(ftle[succ]):+.4f} vs failure {np.median(ftle[~succ]):+.4f}\")\n", + "print(f\"AUC(-λ1 -> success) = {auc(-ftle, succ.astype(int)):.3f} (>0.5 means failures are more chaotic)\")\n", + "plt.figure(figsize=(6,4))\n", + "b=np.linspace(ftle.min(), ftle.max(), 40)\n", + "plt.hist(ftle[succ], b, alpha=.6, color='g', density=True, label=f'success (n={succ.sum()})')\n", + "plt.hist(ftle[~succ], b, alpha=.6, color='r', density=True, label=f'failure (n={(~succ).sum()})')\n", + "plt.axvline(0, ls=':', c='k', lw=1)\n", + "plt.xlabel('leading finite-time Lyapunov exponent λ1'); plt.ylabel('density')\n", + "plt.title(f'{MODEL}: failures are more chaotic'); plt.legend(); plt.tight_layout(); plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "25aa2620", + "metadata": {}, + "source": [ + "## 3. Why — transient chaos: failures *escape* with more compute\n", + "\n", + "Run the recurrence `N_SEG` segments (far past the 16-segment budget) and watch the fate of\n", + "trajectories that fail at segment 16. **TRM** failures escape the chaotic transient and resolve to\n", + "the correct answer; **HRM** failures are far more strongly trapped. Re-run cell 1 with\n", + "`MODEL=\"hrm_sudoku\"` (restart kernel) to compare." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "106f13d3", "metadata": {}, "outputs": [], "source": [ "def extended_rollout(inp, lab, pid, n=256, n_seg=128, seed=0):\n", " rng=np.random.default_rng(seed); idx=rng.choice(len(inp), n, replace=False)\n", " pe=inner.puzzle_emb_len; sf=inner.config.seq_len+pe; hid=inner.config.hidden_size\n", - " is_hrm = hasattr(inner, \"H_level\")\n", + " is_hrm=hasattr(inner, \"H_level\") and getattr(inner,\"H_level\",None) is not None\n", + " Hmod=inner.H_level if is_hrm else inner.L_level\n", " X=torch.tensor(inp[idx].astype(np.int32),device=dev); Y=torch.tensor(lab[idx].astype(np.int32),device=dev)\n", " P=torch.tensor(pid[idx].astype(np.int32),device=dev)\n", " EX=[]; DR=[]\n", @@ -175,11 +209,11 @@ " zL=inner.L_init.unsqueeze(0).expand(n,sf,hid).clone().to(inner.forward_dtype)\n", " si=dict(cos_sin=inner.rotary_emb() if hasattr(inner,\"rotary_emb\") else None)\n", " emb=inner._input_embeddings(X,P); m=Y>0; prev=None\n", - " for _ in range(n_seg):\n", + " for _ in tqdm(range(n_seg), desc=\"rollout (segments)\"):\n", " for _h in range(inner.config.H_cycles):\n", " for _l in range(inner.config.L_cycles):\n", " zL=inner.L_level(zL, zH+emb, **si)\n", - " zH=(inner.H_level if is_hrm else inner.L_level)(zH, zL, **si)\n", + " zH=Hmod(zH, zL, **si)\n", " p=inner.lm_head(zH)[:,pe:].float().argmax(-1)\n", " EX.append(((p==Y)|~m).all(-1).float().cpu().numpy())\n", " DR.append((torch.zeros(n) if prev is None else (zH-prev).float().flatten(1).norm(1).cpu()).numpy())\n", @@ -189,98 +223,95 @@ "ex, dr = extended_rollout(inp, lab, pid, n=256, n_seg=128)\n", "T=ex.shape[1]; fail=ex[:,15]==0; nf=fail.sum()\n", "print(f\"accuracy @16={ex[:,15].mean():.3f} @{T}={ex[:,-1].mean():.3f}\")\n", - "print(f\"of {nf} step-16 failures: self-resolve to CORRECT by seg{T}: {(fail&(ex[:,-1]==1)).sum()/nf*100:.0f}%\")\n", - "ld=dr[:,-4:].mean(1)\n", - "print(f\"median latent drift -- failures {np.median(ld[fail]):.1f} vs successes {np.median(ld[ex[:,15]==1]):.1f}\")\n", + "print(f\"of {nf} step-16 failures: self-resolve to CORRECT by seg{T}: {(fail&(ex[:,-1]==1)).sum()/max(nf,1)*100:.0f}%\")\n", "fig,ax=plt.subplots(1,2,figsize=(11,4))\n", "ax[0].plot(range(1,T+1), ex.mean(0)); ax[0].axvline(16,ls='--',c='gray'); ax[0].set_xscale('log')\n", "ax[0].set_xlabel('inference segments'); ax[0].set_ylabel('accuracy'); ax[0].set_title('accuracy vs compute')\n", - "S=[(fail&(ex[:,:s].max(1)==0)).sum()/nf for s in range(16,T+1)]\n", + "S=[(fail&(ex[:,:s].max(1)==0)).sum()/max(nf,1) for s in range(16,T+1)]\n", "ax[1].plot(range(16,T+1),S); ax[1].set_yscale('log'); ax[1].set_xlabel('segments'); ax[1].set_ylabel('frac failures still unsolved')\n", "ax[1].set_title('escape from chaotic set (straight line on log-y = exponential escape)'); plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", - "id": "912eefb8", + "id": "306d11c9", "metadata": {}, "source": [ "## 4. Basin accessibility — input-determined or initial-condition-determined?\n", "\n", - "The puzzle is re-injected at *every* segment (`z_H + input_embeddings`), so perturbing only the\n", - "**initial** latent state `z0` is a clean initial-condition change that leaves the input intact.\n", - "Restart each step-16 failure `K` times from `z0 + sigma*noise`: if a small kick frees it (some\n", - "restart solves), the solution basin is large and accessible — *initial-condition-determined*; if no\n", - "nearby IC escapes, the trapping is *input-determined*. TRM: a small kick frees most. HRM: a hard\n", - "core escapes no nearby IC. (GPU: seconds. Laptop/CPU with TRM: a couple of minutes — lower `n`/`K`.)" + "The puzzle is re-injected every segment (`z_H + input_embeddings`), so perturbing only the\n", + "**initial** latent `z0` is a clean initial-condition change that leaves the input intact. Restart\n", + "each step-16 failure `K` times from `z0 + sigma*noise`: if a small kick frees it, the solution basin\n", + "is large and accessible (TRM); if no nearby IC escapes, the trapping is input-determined (HRM has a\n", + "hard core). (GPU: seconds. Laptop/CPU with TRM: a couple of minutes — lower `n`/`K`.)" ] }, { "cell_type": "code", "execution_count": null, - "id": "b812488b", + "id": "682927dd", "metadata": {}, "outputs": [], "source": [ "def perturb_z0(inp, lab, pid, n=96, K=8, sigmas=(0.0, 0.1, 0.3, 1.0), n_seg=48, readout=16, seed=0):\n", - " rng = np.random.default_rng(seed); idx = rng.choice(len(inp), n, replace=False)\n", - " pe = inner.puzzle_emb_len; sf = inner.config.seq_len + pe; hid = inner.config.hidden_size\n", - " is_hrm = hasattr(inner, \"H_level\") and getattr(inner, \"H_level\", None) is not None\n", - " Hup = inner.H_level if is_hrm else inner.L_level # weight-tied TRM reuses L_level\n", - " sc = float(inner.H_init.float().std()); g = torch.Generator(device=dev).manual_seed(seed)\n", - " X = torch.tensor(inp[idx].astype(np.int32), device=dev); Y = torch.tensor(lab[idx].astype(np.int32), device=dev)\n", - " P = torch.tensor(pid[idx].astype(np.int32), device=dev)\n", - " si = dict(cos_sin=inner.rotary_emb() if hasattr(inner, \"rotary_emb\") else None)\n", - " solve = np.zeros((n, len(sigmas), K), bool); base = None\n", + " rng=np.random.default_rng(seed); idx=rng.choice(len(inp), n, replace=False)\n", + " pe=inner.puzzle_emb_len; sf=inner.config.seq_len+pe; hid=inner.config.hidden_size\n", + " is_hrm=hasattr(inner,\"H_level\") and getattr(inner,\"H_level\",None) is not None\n", + " Hmod=inner.H_level if is_hrm else inner.L_level\n", + " sc=float(inner.H_init.float().std()); g=torch.Generator(device=dev).manual_seed(seed)\n", + " X=torch.tensor(inp[idx].astype(np.int32),device=dev); Y=torch.tensor(lab[idx].astype(np.int32),device=dev)\n", + " P=torch.tensor(pid[idx].astype(np.int32),device=dev)\n", + " si=dict(cos_sin=inner.rotary_emb() if hasattr(inner,\"rotary_emb\") else None)\n", + " solve=np.zeros((n,len(sigmas),K),bool); base=None\n", " with torch.no_grad():\n", - " emb0 = inner._input_embeddings(X, P); m0 = Y > 0\n", - " for sj, sg in enumerate(sigmas):\n", - " emb = emb0.repeat_interleave(K, 0); Yr = Y.repeat_interleave(K, 0); mr = m0.repeat_interleave(K, 0); B = n * K\n", - " zH = inner.H_init.unsqueeze(0).expand(B, sf, hid).clone().to(inner.forward_dtype)\n", - " zL = inner.L_init.unsqueeze(0).expand(B, sf, hid).clone().to(inner.forward_dtype)\n", - " if sg > 0:\n", - " zH = (zH.float() + sg*sc*torch.randn(zH.shape, generator=g, device=dev)).to(inner.forward_dtype)\n", - " zL = (zL.float() + sg*sc*torch.randn(zL.shape, generator=g, device=dev)).to(inner.forward_dtype)\n", - " solved = torch.zeros(B, dtype=torch.bool, device=dev)\n", + " emb0=inner._input_embeddings(X,P); m0=Y>0\n", + " for sj,sg in enumerate(tqdm(sigmas, desc=\"basin (sigma levels)\")):\n", + " emb=emb0.repeat_interleave(K,0); Yr=Y.repeat_interleave(K,0); mr=m0.repeat_interleave(K,0); B=n*K\n", + " zH=inner.H_init.unsqueeze(0).expand(B,sf,hid).clone().to(inner.forward_dtype)\n", + " zL=inner.L_init.unsqueeze(0).expand(B,sf,hid).clone().to(inner.forward_dtype)\n", + " if sg>0:\n", + " zH=(zH.float()+sg*sc*torch.randn(zH.shape,generator=g,device=dev)).to(inner.forward_dtype)\n", + " zL=(zL.float()+sg*sc*torch.randn(zL.shape,generator=g,device=dev)).to(inner.forward_dtype)\n", + " solved=torch.zeros(B,dtype=torch.bool,device=dev)\n", " for s in range(n_seg):\n", " for _h in range(inner.config.H_cycles):\n", - " for _l in range(inner.config.L_cycles): zL = inner.L_level(zL, zH + emb, **si)\n", - " zH = Hup(zH, zL, **si)\n", - " ok = ((inner.lm_head(zH)[:, pe:].float().argmax(-1) == Yr) | ~mr).all(-1); solved |= ok\n", - " if sj == 0 and s == readout - 1: base = ok.view(n, K)[:, 0].cpu().numpy()\n", - " solve[:, sj] = solved.view(n, K).cpu().numpy()\n", + " for _l in range(inner.config.L_cycles): zL=inner.L_level(zL,zH+emb,**si)\n", + " zH=Hmod(zH,zL,**si)\n", + " ok=((inner.lm_head(zH)[:,pe:].float().argmax(-1)==Yr)|~mr).all(-1); solved|=ok\n", + " if sj==0 and s==readout-1: base=ok.view(n,K)[:,0].cpu().numpy()\n", + " solve[:,sj]=solved.view(n,K).cpu().numpy()\n", " return solve, base, np.array(sigmas)\n", "\n", "solve, base, sg = perturb_z0(inp, lab, pid)\n", - "fail = ~base; nf = int(fail.sum())\n", - "print(f\"{nf} of {len(base)} puzzles fail@{16}; freeing them by restarting from a perturbed IC:\")\n", - "for j, s in enumerate(sg):\n", - " sub = solve[fail, j]; print(f\" sigma={s:.1f}: single-restart={sub.mean():.2f} best-of-K={sub.any(1).mean():.2f}\")\n", - "plt.figure(figsize=(6, 4))\n", - "plt.plot(sg, [solve[fail, j].mean() for j in range(len(sg))], 'o--', label='single restart')\n", - "plt.plot(sg, [solve[fail, j].any(1).mean() for j in range(len(sg))], 's-', label='best-of-K')\n", + "fail=~base; nf=int(fail.sum())\n", + "print(f\"{nf} of {len(base)} puzzles fail@16; freeing them by restarting from a perturbed IC:\")\n", + "for j,s in enumerate(sg):\n", + " sub=solve[fail,j]; print(f\" sigma={s:.1f}: single-restart={sub.mean():.2f} best-of-K={sub.any(1).mean():.2f}\")\n", + "plt.figure(figsize=(6,4))\n", + "plt.plot(sg,[solve[fail,j].mean() for j in range(len(sg))],'o--',label='single restart')\n", + "plt.plot(sg,[solve[fail,j].any(1).mean() for j in range(len(sg))],'s-',label='best-of-K')\n", "plt.xlabel('relative IC noise sigma'); plt.ylabel('solve rate (failing puzzles)')\n", "plt.title('basin accessibility: does a restart free a trapped puzzle?'); plt.legend(); plt.grid(alpha=.3); plt.show()" ] }, { "cell_type": "markdown", - "id": "4e2c8f69", + "id": "22ef7f6f", "metadata": {}, "source": [ "## What this shows\n", - "- **TRM**: step-16 failures *escape* the chaotic transient and resolve to the correct answer\n", - " (≈0 settle to a wrong answer) → a chaotic **saddle** + one solution fixed point. More compute\n", - " solves more puzzles.\n", - "- **HRM**: failures escape too, but *much* more slowly — most are still churning at this horizon.\n", - " Out to 4000 segments the never-correct fraction keeps decaying (≈0.87→0.77), so it is a\n", - " **strongly-trapping chaotic saddle**, NOT a strict attractor. And the per-segment escape-rate gap\n", - " (~5×) is mostly compute-per-segment: TRM evaluates its recurrent module 21×/segment vs HRM 6×, so\n", - " per module-evaluation the gap is only ~1.6×.\n", - "- **Neither settles to a wrong fixed point** — the \"spurious fixed point\" reading from 2D PCA is an\n", - " artifact of projecting high-dimensional chaotic wandering onto two axes.\n", + "- **The result (cell 2):** in the same trained network, failed trajectories have a higher leading\n", + " finite-time Lyapunov exponent than successful ones — failure is locally more chaotic.\n", + "- **Why (cell 3):** that chaos is a *transient*. Failures sit on a chaotic **saddle**, not a wrong\n", + " fixed point — TRM's escape and self-correct with more compute; HRM's are much more strongly\n", + " trapped (still a saddle, just a far smaller escape rate). The per-segment escape gap is mostly\n", + " compute-per-segment (TRM evaluates its module 21×/segment vs HRM 6×; per module-eval the gap is\n", + " only ~1.6×). The \"spurious fixed point\" reading from 2D PCA is an artifact of projecting\n", + " high-dimensional chaotic wandering.\n", + "- **Basin (cell 4):** a small initial-condition kick frees most of TRM's trapped puzzles\n", + " (IC-determined, large basin); a hard core of HRM's escapes no nearby IC (input-determined).\n", "\n", - "Try: change `MODEL`, `N_SEG`, `eps` (toy); compare TRM vs HRM escape curves." + "Try: change `MODEL` (restart kernel), `n`/`n_seg`, and compare TRM vs HRM at every step." ] } ], |